Toniolo and Linder, Equation (3b), real

Percentage Accurate: 93.5% → 99.7%

Time: 13.1s

Alternatives: 22

Speedup: 1.4×

Specification

Math

\[\begin{array}{l} \\ \frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (* (/ (sin ky) (sqrt (+ (pow (sin kx) 2.0) (pow (sin ky) 2.0)))) (sin th)))

C

double code(double kx, double ky, double th) {
	return (sin(ky) / sqrt((pow(sin(kx), 2.0) + pow(sin(ky), 2.0)))) * sin(th);
}

Fortran

real(8) function code(kx, ky, th)
    real(8), intent (in) :: kx
    real(8), intent (in) :: ky
    real(8), intent (in) :: th
    code = (sin(ky) / sqrt(((sin(kx) ** 2.0d0) + (sin(ky) ** 2.0d0)))) * sin(th)
end function

Java

public static double code(double kx, double ky, double th) {
	return (Math.sin(ky) / Math.sqrt((Math.pow(Math.sin(kx), 2.0) + Math.pow(Math.sin(ky), 2.0)))) * Math.sin(th);
}

Python

def code(kx, ky, th):
	return (math.sin(ky) / math.sqrt((math.pow(math.sin(kx), 2.0) + math.pow(math.sin(ky), 2.0)))) * math.sin(th)

Julia

function code(kx, ky, th)
	return Float64(Float64(sin(ky) / sqrt(Float64((sin(kx) ^ 2.0) + (sin(ky) ^ 2.0)))) * sin(th))
end

MATLAB

function tmp = code(kx, ky, th)
	tmp = (sin(ky) / sqrt(((sin(kx) ^ 2.0) + (sin(ky) ^ 2.0)))) * sin(th);
end

Wolfram

code[kx_, ky_, th_] := N[(N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[Sin[ky], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision]

Initial Program: 93.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (* (/ (sin ky) (sqrt (+ (pow (sin kx) 2.0) (pow (sin ky) 2.0)))) (sin th)))

C

double code(double kx, double ky, double th) {
	return (sin(ky) / sqrt((pow(sin(kx), 2.0) + pow(sin(ky), 2.0)))) * sin(th);
}

Fortran

real(8) function code(kx, ky, th)
    real(8), intent (in) :: kx
    real(8), intent (in) :: ky
    real(8), intent (in) :: th
    code = (sin(ky) / sqrt(((sin(kx) ** 2.0d0) + (sin(ky) ** 2.0d0)))) * sin(th)
end function

Java

public static double code(double kx, double ky, double th) {
	return (Math.sin(ky) / Math.sqrt((Math.pow(Math.sin(kx), 2.0) + Math.pow(Math.sin(ky), 2.0)))) * Math.sin(th);
}

Python

def code(kx, ky, th):
	return (math.sin(ky) / math.sqrt((math.pow(math.sin(kx), 2.0) + math.pow(math.sin(ky), 2.0)))) * math.sin(th)

Julia

function code(kx, ky, th)
	return Float64(Float64(sin(ky) / sqrt(Float64((sin(kx) ^ 2.0) + (sin(ky) ^ 2.0)))) * sin(th))
end

MATLAB

function tmp = code(kx, ky, th)
	tmp = (sin(ky) / sqrt(((sin(kx) ^ 2.0) + (sin(ky) ^ 2.0)))) * sin(th);
end

Wolfram

code[kx_, ky_, th_] := N[(N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[Sin[ky], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision]

Alternative 1: 99.7% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \frac{\sin ky}{\mathsf{hypot}\left(\sin kx, \sin ky\right)} \cdot \sin th \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (* (/ (sin ky) (hypot (sin kx) (sin ky))) (sin th)))

C

double code(double kx, double ky, double th) {
	return (sin(ky) / hypot(sin(kx), sin(ky))) * sin(th);
}

Java

public static double code(double kx, double ky, double th) {
	return (Math.sin(ky) / Math.hypot(Math.sin(kx), Math.sin(ky))) * Math.sin(th);
}

Python

def code(kx, ky, th):
	return (math.sin(ky) / math.hypot(math.sin(kx), math.sin(ky))) * math.sin(th)

Julia

function code(kx, ky, th)
	return Float64(Float64(sin(ky) / hypot(sin(kx), sin(ky))) * sin(th))
end

MATLAB

function tmp = code(kx, ky, th)
	tmp = (sin(ky) / hypot(sin(kx), sin(ky))) * sin(th);
end

Wolfram

code[kx_, ky_, th_] := N[(N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[Sin[kx], $MachinePrecision] ^ 2 + N[Sin[ky], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 93.7%

   \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
2. Add Preprocessing

3. Taylor expanded in kx around inf

   \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sin.f64}\left(ky\right), \color{blue}{\left(\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}\right)}\right), \mathsf{sin.f64}\left(th\right)\right) \]
4. Step-by-step derivation

   1. unpow2 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sin.f64}\left(ky\right), \left(\sqrt{\sin kx \cdot \sin kx + {\sin ky}^{2}}\right)\right), \mathsf{sin.f64}\left(th\right)\right) \]

   2. unpow2 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sin.f64}\left(ky\right), \left(\sqrt{\sin kx \cdot \sin kx + \sin ky \cdot \sin ky}\right)\right), \mathsf{sin.f64}\left(th\right)\right) \]

   3. hypot-define N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sin.f64}\left(ky\right), \left(\mathsf{hypot}\left(\sin kx, \sin ky\right)\right)\right), \mathsf{sin.f64}\left(th\right)\right) \]

   4. hypot-lowering-hypot.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{hypot.f64}\left(\sin kx, \sin ky\right)\right), \mathsf{sin.f64}\left(th\right)\right) \]

   5. sin-lowering-sin.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{hypot.f64}\left(\mathsf{sin.f64}\left(kx\right), \sin ky\right)\right), \mathsf{sin.f64}\left(th\right)\right) \]

   6. sin-lowering-sin.f64 99.7%

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{hypot.f64}\left(\mathsf{sin.f64}\left(kx\right), \mathsf{sin.f64}\left(ky\right)\right)\right), \mathsf{sin.f64}\left(th\right)\right) \]

5. Simplified 99.7%

   \[\leadsto \frac{\sin ky}{\color{blue}{\mathsf{hypot}\left(\sin kx, \sin ky\right)}} \cdot \sin th \]
6. Add Preprocessing

Alternative 2: 71.2% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sin ky \leq -0.03:\\ \;\;\;\;\frac{th}{\frac{\mathsf{hypot}\left(\sin ky, kx\right)}{\sin ky}}\\ \mathbf{elif}\;\sin ky \leq 4 \cdot 10^{-7}:\\ \;\;\;\;\frac{\sin th}{\mathsf{hypot}\left(ky, \sin kx\right)} \cdot \left(ky \cdot \left(1 + -0.16666666666666666 \cdot \left(ky \cdot ky\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= (sin ky) -0.03)
   (/ th (/ (hypot (sin ky) kx) (sin ky)))
   (if (<= (sin ky) 4e-7)
     (*
      (/ (sin th) (hypot ky (sin kx)))
      (* ky (+ 1.0 (* -0.16666666666666666 (* ky ky)))))
     (sin th))))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (sin(ky) <= -0.03) {
		tmp = th / (hypot(sin(ky), kx) / sin(ky));
	} else if (sin(ky) <= 4e-7) {
		tmp = (sin(th) / hypot(ky, sin(kx))) * (ky * (1.0 + (-0.16666666666666666 * (ky * ky))));
	} else {
		tmp = sin(th);
	}
	return tmp;
}

Java

public static double code(double kx, double ky, double th) {
	double tmp;
	if (Math.sin(ky) <= -0.03) {
		tmp = th / (Math.hypot(Math.sin(ky), kx) / Math.sin(ky));
	} else if (Math.sin(ky) <= 4e-7) {
		tmp = (Math.sin(th) / Math.hypot(ky, Math.sin(kx))) * (ky * (1.0 + (-0.16666666666666666 * (ky * ky))));
	} else {
		tmp = Math.sin(th);
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if math.sin(ky) <= -0.03:
		tmp = th / (math.hypot(math.sin(ky), kx) / math.sin(ky))
	elif math.sin(ky) <= 4e-7:
		tmp = (math.sin(th) / math.hypot(ky, math.sin(kx))) * (ky * (1.0 + (-0.16666666666666666 * (ky * ky))))
	else:
		tmp = math.sin(th)
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (sin(ky) <= -0.03)
		tmp = Float64(th / Float64(hypot(sin(ky), kx) / sin(ky)));
	elseif (sin(ky) <= 4e-7)
		tmp = Float64(Float64(sin(th) / hypot(ky, sin(kx))) * Float64(ky * Float64(1.0 + Float64(-0.16666666666666666 * Float64(ky * ky)))));
	else
		tmp = sin(th);
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if (sin(ky) <= -0.03)
		tmp = th / (hypot(sin(ky), kx) / sin(ky));
	elseif (sin(ky) <= 4e-7)
		tmp = (sin(th) / hypot(ky, sin(kx))) * (ky * (1.0 + (-0.16666666666666666 * (ky * ky))));
	else
		tmp = sin(th);
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[N[Sin[ky], $MachinePrecision], -0.03], N[(th / N[(N[Sqrt[N[Sin[ky], $MachinePrecision] ^ 2 + kx ^ 2], $MachinePrecision] / N[Sin[ky], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Sin[ky], $MachinePrecision], 4e-7], N[(N[(N[Sin[th], $MachinePrecision] / N[Sqrt[ky ^ 2 + N[Sin[kx], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision] * N[(ky * N[(1.0 + N[(-0.16666666666666666 * N[(ky * ky), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Sin[th], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (sin.f64 ky) < -0.029999999999999999

   1. Initial program 99.7%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing

   3. Taylor expanded in kx around inf

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sin.f64}\left(ky\right), \color{blue}{\left(\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}\right)}\right), \mathsf{sin.f64}\left(th\right)\right) \]
   4. Step-by-step derivation

      1. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sin.f64}\left(ky\right), \left(\sqrt{\sin kx \cdot \sin kx + {\sin ky}^{2}}\right)\right), \mathsf{sin.f64}\left(th\right)\right) \]

      2. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sin.f64}\left(ky\right), \left(\sqrt{\sin kx \cdot \sin kx + \sin ky \cdot \sin ky}\right)\right), \mathsf{sin.f64}\left(th\right)\right) \]

      3. hypot-define N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sin.f64}\left(ky\right), \left(\mathsf{hypot}\left(\sin kx, \sin ky\right)\right)\right), \mathsf{sin.f64}\left(th\right)\right) \]

      4. hypot-lowering-hypot.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{hypot.f64}\left(\sin kx, \sin ky\right)\right), \mathsf{sin.f64}\left(th\right)\right) \]

      5. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{hypot.f64}\left(\mathsf{sin.f64}\left(kx\right), \sin ky\right)\right), \mathsf{sin.f64}\left(th\right)\right) \]

      6. sin-lowering-sin.f64 99.7%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{hypot.f64}\left(\mathsf{sin.f64}\left(kx\right), \mathsf{sin.f64}\left(ky\right)\right)\right), \mathsf{sin.f64}\left(th\right)\right) \]

   5. Simplified 99.7%

      \[\leadsto \frac{\sin ky}{\color{blue}{\mathsf{hypot}\left(\sin kx, \sin ky\right)}} \cdot \sin th \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \sin th \cdot \color{blue}{\frac{\sin ky}{\sqrt{\sin kx \cdot \sin kx + \sin ky \cdot \sin ky}}} \]

      2. clear-num N/A

         \[\leadsto \sin th \cdot \frac{1}{\color{blue}{\frac{\sqrt{\sin kx \cdot \sin kx + \sin ky \cdot \sin ky}}{\sin ky}}} \]

      3. un-div-inv N/A

         \[\leadsto \frac{\sin th}{\color{blue}{\frac{\sqrt{\sin kx \cdot \sin kx + \sin ky \cdot \sin ky}}{\sin ky}}} \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\sin th, \color{blue}{\left(\frac{\sqrt{\sin kx \cdot \sin kx + \sin ky \cdot \sin ky}}{\sin ky}\right)}\right) \]

      5. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sin.f64}\left(th\right), \left(\frac{\color{blue}{\sqrt{\sin kx \cdot \sin kx + \sin ky \cdot \sin ky}}}{\sin ky}\right)\right) \]

      6. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sin.f64}\left(th\right), \left(\frac{\sqrt{\sin ky \cdot \sin ky + \sin kx \cdot \sin kx}}{\sin ky}\right)\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sin.f64}\left(th\right), \mathsf{/.f64}\left(\left(\sqrt{\sin ky \cdot \sin ky + \sin kx \cdot \sin kx}\right), \color{blue}{\sin ky}\right)\right) \]

      8. hypot-define N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sin.f64}\left(th\right), \mathsf{/.f64}\left(\left(\mathsf{hypot}\left(\sin ky, \sin kx\right)\right), \sin \color{blue}{ky}\right)\right) \]

      9. hypot-lowering-hypot.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sin.f64}\left(th\right), \mathsf{/.f64}\left(\mathsf{hypot.f64}\left(\sin ky, \sin kx\right), \sin \color{blue}{ky}\right)\right) \]

      10. sin-lowering-sin.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{sin.f64}\left(th\right), \mathsf{/.f64}\left(\mathsf{hypot.f64}\left(\mathsf{sin.f64}\left(ky\right), \sin kx\right), \sin ky\right)\right) \]

      11. sin-lowering-sin.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{sin.f64}\left(th\right), \mathsf{/.f64}\left(\mathsf{hypot.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{sin.f64}\left(kx\right)\right), \sin ky\right)\right) \]

      12. sin-lowering-sin.f64 99.6%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{sin.f64}\left(th\right), \mathsf{/.f64}\left(\mathsf{hypot.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{sin.f64}\left(kx\right)\right), \mathsf{sin.f64}\left(ky\right)\right)\right) \]

   7. Applied egg-rr 99.6%

      \[\leadsto \color{blue}{\frac{\sin th}{\frac{\mathsf{hypot}\left(\sin ky, \sin kx\right)}{\sin ky}}} \]

   8. Taylor expanded in th around 0

      \[\leadsto \mathsf{/.f64}\left(\color{blue}{th}, \mathsf{/.f64}\left(\mathsf{hypot.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{sin.f64}\left(kx\right)\right), \mathsf{sin.f64}\left(ky\right)\right)\right) \]
   9. Step-by-step derivation

   10. Simplified 47.6%

       \[\leadsto \frac{\color{blue}{th}}{\frac{\mathsf{hypot}\left(\sin ky, \sin kx\right)}{\sin ky}} \]

   11. Taylor expanded in kx around 0

       \[\leadsto \mathsf{/.f64}\left(th, \mathsf{/.f64}\left(\mathsf{hypot.f64}\left(\mathsf{sin.f64}\left(ky\right), \color{blue}{kx}\right), \mathsf{sin.f64}\left(ky\right)\right)\right) \]
   12. Step-by-step derivation

   13. Simplified 23.2%

       \[\leadsto \frac{th}{\frac{\mathsf{hypot}\left(\sin ky, \color{blue}{kx}\right)}{\sin ky}} \]

   if -0.029999999999999999 < (sin.f64 ky) < 3.9999999999999998e-7

   1. Initial program 87.2%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Step-by-step derivation

      1. associate-*l/ N/A

         \[\leadsto \frac{\sin ky \cdot \sin th}{\color{blue}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}} \]

      2. associate-/l* N/A

         \[\leadsto \sin ky \cdot \color{blue}{\frac{\sin th}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\sin ky, \color{blue}{\left(\frac{\sin th}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}\right)}\right) \]

      4. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \left(\frac{\color{blue}{\sin th}}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}\right)\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\sin th, \color{blue}{\left(\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}\right)}\right)\right) \]

      6. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\mathsf{sin.f64}\left(th\right), \left(\sqrt{\color{blue}{{\sin kx}^{2} + {\sin ky}^{2}}}\right)\right)\right) \]

      7. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\mathsf{sin.f64}\left(th\right), \left(\sqrt{{\sin ky}^{2} + {\sin kx}^{2}}\right)\right)\right) \]

      8. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\mathsf{sin.f64}\left(th\right), \left(\sqrt{\sin ky \cdot \sin ky + {\sin kx}^{2}}\right)\right)\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\mathsf{sin.f64}\left(th\right), \left(\sqrt{\sin ky \cdot \sin ky + \sin kx \cdot \sin kx}\right)\right)\right) \]

      10. hypot-define N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\mathsf{sin.f64}\left(th\right), \left(\mathsf{hypot}\left(\sin ky, \color{blue}{\sin kx}\right)\right)\right)\right) \]

      11. hypot-lowering-hypot.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\mathsf{sin.f64}\left(th\right), \mathsf{hypot.f64}\left(\sin ky, \color{blue}{\sin kx}\right)\right)\right) \]

      12. sin-lowering-sin.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\mathsf{sin.f64}\left(th\right), \mathsf{hypot.f64}\left(\mathsf{sin.f64}\left(ky\right), \sin \color{blue}{kx}\right)\right)\right) \]

      13. sin-lowering-sin.f64 99.6%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\mathsf{sin.f64}\left(th\right), \mathsf{hypot.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{sin.f64}\left(kx\right)\right)\right)\right) \]

   3. Simplified 99.6%

      \[\leadsto \color{blue}{\sin ky \cdot \frac{\sin th}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in ky around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\mathsf{sin.f64}\left(th\right), \mathsf{hypot.f64}\left(\color{blue}{ky}, \mathsf{sin.f64}\left(kx\right)\right)\right)\right) \]
   6. Step-by-step derivation

   7. Simplified 99.6%

      \[\leadsto \sin ky \cdot \frac{\sin th}{\mathsf{hypot}\left(\color{blue}{ky}, \sin kx\right)} \]

   8. Taylor expanded in ky around 0

      \[\leadsto \mathsf{*.f64}\left(\color{blue}{\left(ky \cdot \left(1 + \frac{-1}{6} \cdot {ky}^{2}\right)\right)}, \mathsf{/.f64}\left(\mathsf{sin.f64}\left(th\right), \mathsf{hypot.f64}\left(ky, \mathsf{sin.f64}\left(kx\right)\right)\right)\right) \]
   9. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(ky, \left(1 + \frac{-1}{6} \cdot {ky}^{2}\right)\right), \mathsf{/.f64}\left(\color{blue}{\mathsf{sin.f64}\left(th\right)}, \mathsf{hypot.f64}\left(ky, \mathsf{sin.f64}\left(kx\right)\right)\right)\right) \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(ky, \mathsf{+.f64}\left(1, \left(\frac{-1}{6} \cdot {ky}^{2}\right)\right)\right), \mathsf{/.f64}\left(\mathsf{sin.f64}\left(th\right), \mathsf{hypot.f64}\left(ky, \mathsf{sin.f64}\left(kx\right)\right)\right)\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(ky, \mathsf{+.f64}\left(1, \left({ky}^{2} \cdot \frac{-1}{6}\right)\right)\right), \mathsf{/.f64}\left(\mathsf{sin.f64}\left(th\right), \mathsf{hypot.f64}\left(ky, \mathsf{sin.f64}\left(kx\right)\right)\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(ky, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left({ky}^{2}\right), \frac{-1}{6}\right)\right)\right), \mathsf{/.f64}\left(\mathsf{sin.f64}\left(th\right), \mathsf{hypot.f64}\left(ky, \mathsf{sin.f64}\left(kx\right)\right)\right)\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(ky, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(ky \cdot ky\right), \frac{-1}{6}\right)\right)\right), \mathsf{/.f64}\left(\mathsf{sin.f64}\left(th\right), \mathsf{hypot.f64}\left(ky, \mathsf{sin.f64}\left(kx\right)\right)\right)\right) \]

      6. *-lowering-*.f64 99.6%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(ky, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(ky, ky\right), \frac{-1}{6}\right)\right)\right), \mathsf{/.f64}\left(\mathsf{sin.f64}\left(th\right), \mathsf{hypot.f64}\left(ky, \mathsf{sin.f64}\left(kx\right)\right)\right)\right) \]

   10. Simplified 99.6%

       \[\leadsto \color{blue}{\left(ky \cdot \left(1 + \left(ky \cdot ky\right) \cdot -0.16666666666666666\right)\right)} \cdot \frac{\sin th}{\mathsf{hypot}\left(ky, \sin kx\right)} \]

   if 3.9999999999999998e-7 < (sin.f64 ky)

   1. Initial program 99.6%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Step-by-step derivation

      1. associate-*l/ N/A

         \[\leadsto \frac{\sin ky \cdot \sin th}{\color{blue}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}} \]

      2. associate-/l* N/A

         \[\leadsto \sin ky \cdot \color{blue}{\frac{\sin th}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\sin ky, \color{blue}{\left(\frac{\sin th}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}\right)}\right) \]

      4. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \left(\frac{\color{blue}{\sin th}}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}\right)\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\sin th, \color{blue}{\left(\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}\right)}\right)\right) \]

      6. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\mathsf{sin.f64}\left(th\right), \left(\sqrt{\color{blue}{{\sin kx}^{2} + {\sin ky}^{2}}}\right)\right)\right) \]

      7. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\mathsf{sin.f64}\left(th\right), \left(\sqrt{{\sin ky}^{2} + {\sin kx}^{2}}\right)\right)\right) \]

      8. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\mathsf{sin.f64}\left(th\right), \left(\sqrt{\sin ky \cdot \sin ky + {\sin kx}^{2}}\right)\right)\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\mathsf{sin.f64}\left(th\right), \left(\sqrt{\sin ky \cdot \sin ky + \sin kx \cdot \sin kx}\right)\right)\right) \]

      10. hypot-define N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\mathsf{sin.f64}\left(th\right), \left(\mathsf{hypot}\left(\sin ky, \color{blue}{\sin kx}\right)\right)\right)\right) \]

      11. hypot-lowering-hypot.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\mathsf{sin.f64}\left(th\right), \mathsf{hypot.f64}\left(\sin ky, \color{blue}{\sin kx}\right)\right)\right) \]

      12. sin-lowering-sin.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\mathsf{sin.f64}\left(th\right), \mathsf{hypot.f64}\left(\mathsf{sin.f64}\left(ky\right), \sin \color{blue}{kx}\right)\right)\right) \]

      13. sin-lowering-sin.f64 99.5%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\mathsf{sin.f64}\left(th\right), \mathsf{hypot.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{sin.f64}\left(kx\right)\right)\right)\right) \]

   3. Simplified 99.5%

      \[\leadsto \color{blue}{\sin ky \cdot \frac{\sin th}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in kx around 0

      \[\leadsto \color{blue}{\sin th} \]
   6. Step-by-step derivation

      1. sin-lowering-sin.f64 60.3%

         \[\leadsto \mathsf{sin.f64}\left(th\right) \]

   7. Simplified 60.3%

      \[\leadsto \color{blue}{\sin th} \]
3. Recombined 3 regimes into one program.

4. Final simplification 68.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sin ky \leq -0.03:\\ \;\;\;\;\frac{th}{\frac{\mathsf{hypot}\left(\sin ky, kx\right)}{\sin ky}}\\ \mathbf{elif}\;\sin ky \leq 4 \cdot 10^{-7}:\\ \;\;\;\;\frac{\sin th}{\mathsf{hypot}\left(ky, \sin kx\right)} \cdot \left(ky \cdot \left(1 + -0.16666666666666666 \cdot \left(ky \cdot ky\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 71.4% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sin ky \leq -0.03:\\ \;\;\;\;\frac{th}{\frac{\mathsf{hypot}\left(\sin ky, kx\right)}{\sin ky}}\\ \mathbf{elif}\;\sin ky \leq 0.02:\\ \;\;\;\;ky \cdot \frac{\sin th}{\mathsf{hypot}\left(ky, \sin kx\right)}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= (sin ky) -0.03)
   (/ th (/ (hypot (sin ky) kx) (sin ky)))
   (if (<= (sin ky) 0.02) (* ky (/ (sin th) (hypot ky (sin kx)))) (sin th))))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (sin(ky) <= -0.03) {
		tmp = th / (hypot(sin(ky), kx) / sin(ky));
	} else if (sin(ky) <= 0.02) {
		tmp = ky * (sin(th) / hypot(ky, sin(kx)));
	} else {
		tmp = sin(th);
	}
	return tmp;
}

Java

public static double code(double kx, double ky, double th) {
	double tmp;
	if (Math.sin(ky) <= -0.03) {
		tmp = th / (Math.hypot(Math.sin(ky), kx) / Math.sin(ky));
	} else if (Math.sin(ky) <= 0.02) {
		tmp = ky * (Math.sin(th) / Math.hypot(ky, Math.sin(kx)));
	} else {
		tmp = Math.sin(th);
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if math.sin(ky) <= -0.03:
		tmp = th / (math.hypot(math.sin(ky), kx) / math.sin(ky))
	elif math.sin(ky) <= 0.02:
		tmp = ky * (math.sin(th) / math.hypot(ky, math.sin(kx)))
	else:
		tmp = math.sin(th)
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (sin(ky) <= -0.03)
		tmp = Float64(th / Float64(hypot(sin(ky), kx) / sin(ky)));
	elseif (sin(ky) <= 0.02)
		tmp = Float64(ky * Float64(sin(th) / hypot(ky, sin(kx))));
	else
		tmp = sin(th);
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if (sin(ky) <= -0.03)
		tmp = th / (hypot(sin(ky), kx) / sin(ky));
	elseif (sin(ky) <= 0.02)
		tmp = ky * (sin(th) / hypot(ky, sin(kx)));
	else
		tmp = sin(th);
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[N[Sin[ky], $MachinePrecision], -0.03], N[(th / N[(N[Sqrt[N[Sin[ky], $MachinePrecision] ^ 2 + kx ^ 2], $MachinePrecision] / N[Sin[ky], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Sin[ky], $MachinePrecision], 0.02], N[(ky * N[(N[Sin[th], $MachinePrecision] / N[Sqrt[ky ^ 2 + N[Sin[kx], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Sin[th], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (sin.f64 ky) < -0.029999999999999999

   1. Initial program 99.7%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing

   3. Taylor expanded in kx around inf

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sin.f64}\left(ky\right), \color{blue}{\left(\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}\right)}\right), \mathsf{sin.f64}\left(th\right)\right) \]
   4. Step-by-step derivation

      1. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sin.f64}\left(ky\right), \left(\sqrt{\sin kx \cdot \sin kx + {\sin ky}^{2}}\right)\right), \mathsf{sin.f64}\left(th\right)\right) \]

      2. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sin.f64}\left(ky\right), \left(\sqrt{\sin kx \cdot \sin kx + \sin ky \cdot \sin ky}\right)\right), \mathsf{sin.f64}\left(th\right)\right) \]

      3. hypot-define N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sin.f64}\left(ky\right), \left(\mathsf{hypot}\left(\sin kx, \sin ky\right)\right)\right), \mathsf{sin.f64}\left(th\right)\right) \]

      4. hypot-lowering-hypot.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{hypot.f64}\left(\sin kx, \sin ky\right)\right), \mathsf{sin.f64}\left(th\right)\right) \]

      5. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{hypot.f64}\left(\mathsf{sin.f64}\left(kx\right), \sin ky\right)\right), \mathsf{sin.f64}\left(th\right)\right) \]

      6. sin-lowering-sin.f64 99.7%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{hypot.f64}\left(\mathsf{sin.f64}\left(kx\right), \mathsf{sin.f64}\left(ky\right)\right)\right), \mathsf{sin.f64}\left(th\right)\right) \]

   5. Simplified 99.7%

      \[\leadsto \frac{\sin ky}{\color{blue}{\mathsf{hypot}\left(\sin kx, \sin ky\right)}} \cdot \sin th \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \sin th \cdot \color{blue}{\frac{\sin ky}{\sqrt{\sin kx \cdot \sin kx + \sin ky \cdot \sin ky}}} \]

      2. clear-num N/A

         \[\leadsto \sin th \cdot \frac{1}{\color{blue}{\frac{\sqrt{\sin kx \cdot \sin kx + \sin ky \cdot \sin ky}}{\sin ky}}} \]

      3. un-div-inv N/A

         \[\leadsto \frac{\sin th}{\color{blue}{\frac{\sqrt{\sin kx \cdot \sin kx + \sin ky \cdot \sin ky}}{\sin ky}}} \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\sin th, \color{blue}{\left(\frac{\sqrt{\sin kx \cdot \sin kx + \sin ky \cdot \sin ky}}{\sin ky}\right)}\right) \]

      5. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sin.f64}\left(th\right), \left(\frac{\color{blue}{\sqrt{\sin kx \cdot \sin kx + \sin ky \cdot \sin ky}}}{\sin ky}\right)\right) \]

      6. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sin.f64}\left(th\right), \left(\frac{\sqrt{\sin ky \cdot \sin ky + \sin kx \cdot \sin kx}}{\sin ky}\right)\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sin.f64}\left(th\right), \mathsf{/.f64}\left(\left(\sqrt{\sin ky \cdot \sin ky + \sin kx \cdot \sin kx}\right), \color{blue}{\sin ky}\right)\right) \]

      8. hypot-define N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sin.f64}\left(th\right), \mathsf{/.f64}\left(\left(\mathsf{hypot}\left(\sin ky, \sin kx\right)\right), \sin \color{blue}{ky}\right)\right) \]

      9. hypot-lowering-hypot.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sin.f64}\left(th\right), \mathsf{/.f64}\left(\mathsf{hypot.f64}\left(\sin ky, \sin kx\right), \sin \color{blue}{ky}\right)\right) \]

      10. sin-lowering-sin.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{sin.f64}\left(th\right), \mathsf{/.f64}\left(\mathsf{hypot.f64}\left(\mathsf{sin.f64}\left(ky\right), \sin kx\right), \sin ky\right)\right) \]

      11. sin-lowering-sin.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{sin.f64}\left(th\right), \mathsf{/.f64}\left(\mathsf{hypot.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{sin.f64}\left(kx\right)\right), \sin ky\right)\right) \]

      12. sin-lowering-sin.f64 99.6%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{sin.f64}\left(th\right), \mathsf{/.f64}\left(\mathsf{hypot.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{sin.f64}\left(kx\right)\right), \mathsf{sin.f64}\left(ky\right)\right)\right) \]

   7. Applied egg-rr 99.6%

      \[\leadsto \color{blue}{\frac{\sin th}{\frac{\mathsf{hypot}\left(\sin ky, \sin kx\right)}{\sin ky}}} \]

   8. Taylor expanded in th around 0

      \[\leadsto \mathsf{/.f64}\left(\color{blue}{th}, \mathsf{/.f64}\left(\mathsf{hypot.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{sin.f64}\left(kx\right)\right), \mathsf{sin.f64}\left(ky\right)\right)\right) \]
   9. Step-by-step derivation

   10. Simplified 47.6%

       \[\leadsto \frac{\color{blue}{th}}{\frac{\mathsf{hypot}\left(\sin ky, \sin kx\right)}{\sin ky}} \]

   11. Taylor expanded in kx around 0

       \[\leadsto \mathsf{/.f64}\left(th, \mathsf{/.f64}\left(\mathsf{hypot.f64}\left(\mathsf{sin.f64}\left(ky\right), \color{blue}{kx}\right), \mathsf{sin.f64}\left(ky\right)\right)\right) \]
   12. Step-by-step derivation

   13. Simplified 23.2%

       \[\leadsto \frac{th}{\frac{\mathsf{hypot}\left(\sin ky, \color{blue}{kx}\right)}{\sin ky}} \]

   if -0.029999999999999999 < (sin.f64 ky) < 0.0200000000000000004

   1. Initial program 87.3%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Step-by-step derivation

      1. associate-*l/ N/A

         \[\leadsto \frac{\sin ky \cdot \sin th}{\color{blue}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}} \]

      2. associate-/l* N/A

         \[\leadsto \sin ky \cdot \color{blue}{\frac{\sin th}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\sin ky, \color{blue}{\left(\frac{\sin th}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}\right)}\right) \]

      4. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \left(\frac{\color{blue}{\sin th}}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}\right)\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\sin th, \color{blue}{\left(\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}\right)}\right)\right) \]

      6. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\mathsf{sin.f64}\left(th\right), \left(\sqrt{\color{blue}{{\sin kx}^{2} + {\sin ky}^{2}}}\right)\right)\right) \]

      7. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\mathsf{sin.f64}\left(th\right), \left(\sqrt{{\sin ky}^{2} + {\sin kx}^{2}}\right)\right)\right) \]

      8. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\mathsf{sin.f64}\left(th\right), \left(\sqrt{\sin ky \cdot \sin ky + {\sin kx}^{2}}\right)\right)\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\mathsf{sin.f64}\left(th\right), \left(\sqrt{\sin ky \cdot \sin ky + \sin kx \cdot \sin kx}\right)\right)\right) \]

      10. hypot-define N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\mathsf{sin.f64}\left(th\right), \left(\mathsf{hypot}\left(\sin ky, \color{blue}{\sin kx}\right)\right)\right)\right) \]

      11. hypot-lowering-hypot.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\mathsf{sin.f64}\left(th\right), \mathsf{hypot.f64}\left(\sin ky, \color{blue}{\sin kx}\right)\right)\right) \]

      12. sin-lowering-sin.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\mathsf{sin.f64}\left(th\right), \mathsf{hypot.f64}\left(\mathsf{sin.f64}\left(ky\right), \sin \color{blue}{kx}\right)\right)\right) \]

      13. sin-lowering-sin.f64 99.6%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\mathsf{sin.f64}\left(th\right), \mathsf{hypot.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{sin.f64}\left(kx\right)\right)\right)\right) \]

   3. Simplified 99.6%

      \[\leadsto \color{blue}{\sin ky \cdot \frac{\sin th}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in ky around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\mathsf{sin.f64}\left(th\right), \mathsf{hypot.f64}\left(\color{blue}{ky}, \mathsf{sin.f64}\left(kx\right)\right)\right)\right) \]
   6. Step-by-step derivation

   7. Simplified 98.9%

      \[\leadsto \sin ky \cdot \frac{\sin th}{\mathsf{hypot}\left(\color{blue}{ky}, \sin kx\right)} \]

   8. Taylor expanded in ky around 0

      \[\leadsto \mathsf{*.f64}\left(\color{blue}{ky}, \mathsf{/.f64}\left(\mathsf{sin.f64}\left(th\right), \mathsf{hypot.f64}\left(ky, \mathsf{sin.f64}\left(kx\right)\right)\right)\right) \]
   9. Step-by-step derivation

   10. Simplified 98.9%

       \[\leadsto \color{blue}{ky} \cdot \frac{\sin th}{\mathsf{hypot}\left(ky, \sin kx\right)} \]

   if 0.0200000000000000004 < (sin.f64 ky)

   1. Initial program 99.6%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Step-by-step derivation

      1. associate-*l/ N/A

         \[\leadsto \frac{\sin ky \cdot \sin th}{\color{blue}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}} \]

      2. associate-/l* N/A

         \[\leadsto \sin ky \cdot \color{blue}{\frac{\sin th}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\sin ky, \color{blue}{\left(\frac{\sin th}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}\right)}\right) \]

      4. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \left(\frac{\color{blue}{\sin th}}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}\right)\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\sin th, \color{blue}{\left(\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}\right)}\right)\right) \]

      6. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\mathsf{sin.f64}\left(th\right), \left(\sqrt{\color{blue}{{\sin kx}^{2} + {\sin ky}^{2}}}\right)\right)\right) \]

      7. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\mathsf{sin.f64}\left(th\right), \left(\sqrt{{\sin ky}^{2} + {\sin kx}^{2}}\right)\right)\right) \]

      8. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\mathsf{sin.f64}\left(th\right), \left(\sqrt{\sin ky \cdot \sin ky + {\sin kx}^{2}}\right)\right)\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\mathsf{sin.f64}\left(th\right), \left(\sqrt{\sin ky \cdot \sin ky + \sin kx \cdot \sin kx}\right)\right)\right) \]

      10. hypot-define N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\mathsf{sin.f64}\left(th\right), \left(\mathsf{hypot}\left(\sin ky, \color{blue}{\sin kx}\right)\right)\right)\right) \]

      11. hypot-lowering-hypot.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\mathsf{sin.f64}\left(th\right), \mathsf{hypot.f64}\left(\sin ky, \color{blue}{\sin kx}\right)\right)\right) \]

      12. sin-lowering-sin.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\mathsf{sin.f64}\left(th\right), \mathsf{hypot.f64}\left(\mathsf{sin.f64}\left(ky\right), \sin \color{blue}{kx}\right)\right)\right) \]

      13. sin-lowering-sin.f64 99.6%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\mathsf{sin.f64}\left(th\right), \mathsf{hypot.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{sin.f64}\left(kx\right)\right)\right)\right) \]

   3. Simplified 99.6%

      \[\leadsto \color{blue}{\sin ky \cdot \frac{\sin th}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in kx around 0

      \[\leadsto \color{blue}{\sin th} \]
   6. Step-by-step derivation

      1. sin-lowering-sin.f64 61.0%

         \[\leadsto \mathsf{sin.f64}\left(th\right) \]

   7. Simplified 61.0%

      \[\leadsto \color{blue}{\sin th} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 4: 71.3% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sin ky \leq -0.03:\\ \;\;\;\;\sin ky \cdot \frac{th}{\mathsf{hypot}\left(\sin ky, kx\right)}\\ \mathbf{elif}\;\sin ky \leq 0.02:\\ \;\;\;\;ky \cdot \frac{\sin th}{\mathsf{hypot}\left(ky, \sin kx\right)}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= (sin ky) -0.03)
   (* (sin ky) (/ th (hypot (sin ky) kx)))
   (if (<= (sin ky) 0.02) (* ky (/ (sin th) (hypot ky (sin kx)))) (sin th))))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (sin(ky) <= -0.03) {
		tmp = sin(ky) * (th / hypot(sin(ky), kx));
	} else if (sin(ky) <= 0.02) {
		tmp = ky * (sin(th) / hypot(ky, sin(kx)));
	} else {
		tmp = sin(th);
	}
	return tmp;
}

Java

public static double code(double kx, double ky, double th) {
	double tmp;
	if (Math.sin(ky) <= -0.03) {
		tmp = Math.sin(ky) * (th / Math.hypot(Math.sin(ky), kx));
	} else if (Math.sin(ky) <= 0.02) {
		tmp = ky * (Math.sin(th) / Math.hypot(ky, Math.sin(kx)));
	} else {
		tmp = Math.sin(th);
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if math.sin(ky) <= -0.03:
		tmp = math.sin(ky) * (th / math.hypot(math.sin(ky), kx))
	elif math.sin(ky) <= 0.02:
		tmp = ky * (math.sin(th) / math.hypot(ky, math.sin(kx)))
	else:
		tmp = math.sin(th)
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (sin(ky) <= -0.03)
		tmp = Float64(sin(ky) * Float64(th / hypot(sin(ky), kx)));
	elseif (sin(ky) <= 0.02)
		tmp = Float64(ky * Float64(sin(th) / hypot(ky, sin(kx))));
	else
		tmp = sin(th);
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if (sin(ky) <= -0.03)
		tmp = sin(ky) * (th / hypot(sin(ky), kx));
	elseif (sin(ky) <= 0.02)
		tmp = ky * (sin(th) / hypot(ky, sin(kx)));
	else
		tmp = sin(th);
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[N[Sin[ky], $MachinePrecision], -0.03], N[(N[Sin[ky], $MachinePrecision] * N[(th / N[Sqrt[N[Sin[ky], $MachinePrecision] ^ 2 + kx ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Sin[ky], $MachinePrecision], 0.02], N[(ky * N[(N[Sin[th], $MachinePrecision] / N[Sqrt[ky ^ 2 + N[Sin[kx], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Sin[th], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (sin.f64 ky) < -0.029999999999999999

   1. Initial program 99.7%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Step-by-step derivation

      1. associate-*l/ N/A

         \[\leadsto \frac{\sin ky \cdot \sin th}{\color{blue}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}} \]

      2. associate-/l* N/A

         \[\leadsto \sin ky \cdot \color{blue}{\frac{\sin th}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\sin ky, \color{blue}{\left(\frac{\sin th}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}\right)}\right) \]

      4. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \left(\frac{\color{blue}{\sin th}}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}\right)\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\sin th, \color{blue}{\left(\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}\right)}\right)\right) \]

      6. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\mathsf{sin.f64}\left(th\right), \left(\sqrt{\color{blue}{{\sin kx}^{2} + {\sin ky}^{2}}}\right)\right)\right) \]

      7. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\mathsf{sin.f64}\left(th\right), \left(\sqrt{{\sin ky}^{2} + {\sin kx}^{2}}\right)\right)\right) \]

      8. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\mathsf{sin.f64}\left(th\right), \left(\sqrt{\sin ky \cdot \sin ky + {\sin kx}^{2}}\right)\right)\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\mathsf{sin.f64}\left(th\right), \left(\sqrt{\sin ky \cdot \sin ky + \sin kx \cdot \sin kx}\right)\right)\right) \]

      10. hypot-define N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\mathsf{sin.f64}\left(th\right), \left(\mathsf{hypot}\left(\sin ky, \color{blue}{\sin kx}\right)\right)\right)\right) \]

      11. hypot-lowering-hypot.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\mathsf{sin.f64}\left(th\right), \mathsf{hypot.f64}\left(\sin ky, \color{blue}{\sin kx}\right)\right)\right) \]

      12. sin-lowering-sin.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\mathsf{sin.f64}\left(th\right), \mathsf{hypot.f64}\left(\mathsf{sin.f64}\left(ky\right), \sin \color{blue}{kx}\right)\right)\right) \]

      13. sin-lowering-sin.f64 99.5%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\mathsf{sin.f64}\left(th\right), \mathsf{hypot.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{sin.f64}\left(kx\right)\right)\right)\right) \]

   3. Simplified 99.5%

      \[\leadsto \color{blue}{\sin ky \cdot \frac{\sin th}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in th around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\color{blue}{\left(th \cdot \left(1 + {th}^{2} \cdot \left(\frac{1}{120} \cdot {th}^{2} - \frac{1}{6}\right)\right)\right)}, \mathsf{hypot.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{sin.f64}\left(kx\right)\right)\right)\right) \]
   6. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(th, \left(1 + {th}^{2} \cdot \left(\frac{1}{120} \cdot {th}^{2} - \frac{1}{6}\right)\right)\right), \mathsf{hypot.f64}\left(\color{blue}{\mathsf{sin.f64}\left(ky\right)}, \mathsf{sin.f64}\left(kx\right)\right)\right)\right) \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(th, \mathsf{+.f64}\left(1, \left({th}^{2} \cdot \left(\frac{1}{120} \cdot {th}^{2} - \frac{1}{6}\right)\right)\right)\right), \mathsf{hypot.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{sin.f64}\left(kx\right)\right)\right)\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(th, \mathsf{+.f64}\left(1, \left(\left(th \cdot th\right) \cdot \left(\frac{1}{120} \cdot {th}^{2} - \frac{1}{6}\right)\right)\right)\right), \mathsf{hypot.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{sin.f64}\left(kx\right)\right)\right)\right) \]

      4. associate-*l* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(th, \mathsf{+.f64}\left(1, \left(th \cdot \left(th \cdot \left(\frac{1}{120} \cdot {th}^{2} - \frac{1}{6}\right)\right)\right)\right)\right), \mathsf{hypot.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{sin.f64}\left(kx\right)\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(th, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(th, \left(th \cdot \left(\frac{1}{120} \cdot {th}^{2} - \frac{1}{6}\right)\right)\right)\right)\right), \mathsf{hypot.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{sin.f64}\left(kx\right)\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(th, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(th, \mathsf{*.f64}\left(th, \left(\frac{1}{120} \cdot {th}^{2} - \frac{1}{6}\right)\right)\right)\right)\right), \mathsf{hypot.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{sin.f64}\left(kx\right)\right)\right)\right) \]

      7. sub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(th, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(th, \mathsf{*.f64}\left(th, \left(\frac{1}{120} \cdot {th}^{2} + \left(\mathsf{neg}\left(\frac{1}{6}\right)\right)\right)\right)\right)\right)\right), \mathsf{hypot.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{sin.f64}\left(kx\right)\right)\right)\right) \]

      8. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(th, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(th, \mathsf{*.f64}\left(th, \left(\frac{1}{120} \cdot {th}^{2} + \frac{-1}{6}\right)\right)\right)\right)\right), \mathsf{hypot.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{sin.f64}\left(kx\right)\right)\right)\right) \]

      9. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(th, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(th, \mathsf{*.f64}\left(th, \left(\frac{-1}{6} + \frac{1}{120} \cdot {th}^{2}\right)\right)\right)\right)\right), \mathsf{hypot.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{sin.f64}\left(kx\right)\right)\right)\right) \]

      10. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(th, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(th, \mathsf{*.f64}\left(th, \mathsf{+.f64}\left(\frac{-1}{6}, \left(\frac{1}{120} \cdot {th}^{2}\right)\right)\right)\right)\right)\right), \mathsf{hypot.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{sin.f64}\left(kx\right)\right)\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(th, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(th, \mathsf{*.f64}\left(th, \mathsf{+.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(\frac{1}{120}, \left({th}^{2}\right)\right)\right)\right)\right)\right)\right), \mathsf{hypot.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{sin.f64}\left(kx\right)\right)\right)\right) \]

      12. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(th, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(th, \mathsf{*.f64}\left(th, \mathsf{+.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(\frac{1}{120}, \left(th \cdot th\right)\right)\right)\right)\right)\right)\right), \mathsf{hypot.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{sin.f64}\left(kx\right)\right)\right)\right) \]

      13. *-lowering-*.f64 47.6%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(th, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(th, \mathsf{*.f64}\left(th, \mathsf{+.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(\frac{1}{120}, \mathsf{*.f64}\left(th, th\right)\right)\right)\right)\right)\right)\right), \mathsf{hypot.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{sin.f64}\left(kx\right)\right)\right)\right) \]

   7. Simplified 47.6%

      \[\leadsto \sin ky \cdot \frac{\color{blue}{th \cdot \left(1 + th \cdot \left(th \cdot \left(-0.16666666666666666 + 0.008333333333333333 \cdot \left(th \cdot th\right)\right)\right)\right)}}{\mathsf{hypot}\left(\sin ky, \sin kx\right)} \]

   8. Taylor expanded in kx around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(th, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(th, \mathsf{*.f64}\left(th, \mathsf{+.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(\frac{1}{120}, \mathsf{*.f64}\left(th, th\right)\right)\right)\right)\right)\right)\right), \mathsf{hypot.f64}\left(\mathsf{sin.f64}\left(ky\right), \color{blue}{kx}\right)\right)\right) \]
   9. Step-by-step derivation

   10. Simplified 22.5%

       \[\leadsto \sin ky \cdot \frac{th \cdot \left(1 + th \cdot \left(th \cdot \left(-0.16666666666666666 + 0.008333333333333333 \cdot \left(th \cdot th\right)\right)\right)\right)}{\mathsf{hypot}\left(\sin ky, \color{blue}{kx}\right)} \]

   11. Taylor expanded in th around 0

       \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\color{blue}{th}, \mathsf{hypot.f64}\left(\mathsf{sin.f64}\left(ky\right), kx\right)\right)\right) \]
   12. Step-by-step derivation

   13. Simplified 23.2%

       \[\leadsto \sin ky \cdot \frac{\color{blue}{th}}{\mathsf{hypot}\left(\sin ky, kx\right)} \]

   if -0.029999999999999999 < (sin.f64 ky) < 0.0200000000000000004

   1. Initial program 87.3%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Step-by-step derivation

      1. associate-*l/ N/A

         \[\leadsto \frac{\sin ky \cdot \sin th}{\color{blue}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}} \]

      2. associate-/l* N/A

         \[\leadsto \sin ky \cdot \color{blue}{\frac{\sin th}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\sin ky, \color{blue}{\left(\frac{\sin th}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}\right)}\right) \]

      4. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \left(\frac{\color{blue}{\sin th}}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}\right)\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\sin th, \color{blue}{\left(\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}\right)}\right)\right) \]

      6. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\mathsf{sin.f64}\left(th\right), \left(\sqrt{\color{blue}{{\sin kx}^{2} + {\sin ky}^{2}}}\right)\right)\right) \]

      7. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\mathsf{sin.f64}\left(th\right), \left(\sqrt{{\sin ky}^{2} + {\sin kx}^{2}}\right)\right)\right) \]

      8. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\mathsf{sin.f64}\left(th\right), \left(\sqrt{\sin ky \cdot \sin ky + {\sin kx}^{2}}\right)\right)\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\mathsf{sin.f64}\left(th\right), \left(\sqrt{\sin ky \cdot \sin ky + \sin kx \cdot \sin kx}\right)\right)\right) \]

      10. hypot-define N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\mathsf{sin.f64}\left(th\right), \left(\mathsf{hypot}\left(\sin ky, \color{blue}{\sin kx}\right)\right)\right)\right) \]

      11. hypot-lowering-hypot.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\mathsf{sin.f64}\left(th\right), \mathsf{hypot.f64}\left(\sin ky, \color{blue}{\sin kx}\right)\right)\right) \]

      12. sin-lowering-sin.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\mathsf{sin.f64}\left(th\right), \mathsf{hypot.f64}\left(\mathsf{sin.f64}\left(ky\right), \sin \color{blue}{kx}\right)\right)\right) \]

      13. sin-lowering-sin.f64 99.6%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\mathsf{sin.f64}\left(th\right), \mathsf{hypot.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{sin.f64}\left(kx\right)\right)\right)\right) \]

   3. Simplified 99.6%

      \[\leadsto \color{blue}{\sin ky \cdot \frac{\sin th}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in ky around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\mathsf{sin.f64}\left(th\right), \mathsf{hypot.f64}\left(\color{blue}{ky}, \mathsf{sin.f64}\left(kx\right)\right)\right)\right) \]
   6. Step-by-step derivation

   7. Simplified 98.9%

      \[\leadsto \sin ky \cdot \frac{\sin th}{\mathsf{hypot}\left(\color{blue}{ky}, \sin kx\right)} \]

   8. Taylor expanded in ky around 0

      \[\leadsto \mathsf{*.f64}\left(\color{blue}{ky}, \mathsf{/.f64}\left(\mathsf{sin.f64}\left(th\right), \mathsf{hypot.f64}\left(ky, \mathsf{sin.f64}\left(kx\right)\right)\right)\right) \]
   9. Step-by-step derivation

   10. Simplified 98.9%

       \[\leadsto \color{blue}{ky} \cdot \frac{\sin th}{\mathsf{hypot}\left(ky, \sin kx\right)} \]

   if 0.0200000000000000004 < (sin.f64 ky)

   1. Initial program 99.6%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Step-by-step derivation

      1. associate-*l/ N/A

         \[\leadsto \frac{\sin ky \cdot \sin th}{\color{blue}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}} \]

      2. associate-/l* N/A

         \[\leadsto \sin ky \cdot \color{blue}{\frac{\sin th}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\sin ky, \color{blue}{\left(\frac{\sin th}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}\right)}\right) \]

      4. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \left(\frac{\color{blue}{\sin th}}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}\right)\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\sin th, \color{blue}{\left(\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}\right)}\right)\right) \]

      6. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\mathsf{sin.f64}\left(th\right), \left(\sqrt{\color{blue}{{\sin kx}^{2} + {\sin ky}^{2}}}\right)\right)\right) \]

      7. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\mathsf{sin.f64}\left(th\right), \left(\sqrt{{\sin ky}^{2} + {\sin kx}^{2}}\right)\right)\right) \]

      8. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\mathsf{sin.f64}\left(th\right), \left(\sqrt{\sin ky \cdot \sin ky + {\sin kx}^{2}}\right)\right)\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\mathsf{sin.f64}\left(th\right), \left(\sqrt{\sin ky \cdot \sin ky + \sin kx \cdot \sin kx}\right)\right)\right) \]

      10. hypot-define N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\mathsf{sin.f64}\left(th\right), \left(\mathsf{hypot}\left(\sin ky, \color{blue}{\sin kx}\right)\right)\right)\right) \]

      11. hypot-lowering-hypot.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\mathsf{sin.f64}\left(th\right), \mathsf{hypot.f64}\left(\sin ky, \color{blue}{\sin kx}\right)\right)\right) \]

      12. sin-lowering-sin.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\mathsf{sin.f64}\left(th\right), \mathsf{hypot.f64}\left(\mathsf{sin.f64}\left(ky\right), \sin \color{blue}{kx}\right)\right)\right) \]

      13. sin-lowering-sin.f64 99.6%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\mathsf{sin.f64}\left(th\right), \mathsf{hypot.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{sin.f64}\left(kx\right)\right)\right)\right) \]

   3. Simplified 99.6%

      \[\leadsto \color{blue}{\sin ky \cdot \frac{\sin th}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in kx around 0

      \[\leadsto \color{blue}{\sin th} \]
   6. Step-by-step derivation

      1. sin-lowering-sin.f64 61.0%

         \[\leadsto \mathsf{sin.f64}\left(th\right) \]

   7. Simplified 61.0%

      \[\leadsto \color{blue}{\sin th} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 5: 66.1% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sin ky \leq -0.02:\\ \;\;\;\;\sin ky \cdot \frac{\sin th}{\sin kx}\\ \mathbf{elif}\;\sin ky \leq 0.02:\\ \;\;\;\;ky \cdot \frac{\sin th}{\mathsf{hypot}\left(ky, \sin kx\right)}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= (sin ky) -0.02)
   (* (sin ky) (/ (sin th) (sin kx)))
   (if (<= (sin ky) 0.02) (* ky (/ (sin th) (hypot ky (sin kx)))) (sin th))))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (sin(ky) <= -0.02) {
		tmp = sin(ky) * (sin(th) / sin(kx));
	} else if (sin(ky) <= 0.02) {
		tmp = ky * (sin(th) / hypot(ky, sin(kx)));
	} else {
		tmp = sin(th);
	}
	return tmp;
}

Java

public static double code(double kx, double ky, double th) {
	double tmp;
	if (Math.sin(ky) <= -0.02) {
		tmp = Math.sin(ky) * (Math.sin(th) / Math.sin(kx));
	} else if (Math.sin(ky) <= 0.02) {
		tmp = ky * (Math.sin(th) / Math.hypot(ky, Math.sin(kx)));
	} else {
		tmp = Math.sin(th);
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if math.sin(ky) <= -0.02:
		tmp = math.sin(ky) * (math.sin(th) / math.sin(kx))
	elif math.sin(ky) <= 0.02:
		tmp = ky * (math.sin(th) / math.hypot(ky, math.sin(kx)))
	else:
		tmp = math.sin(th)
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (sin(ky) <= -0.02)
		tmp = Float64(sin(ky) * Float64(sin(th) / sin(kx)));
	elseif (sin(ky) <= 0.02)
		tmp = Float64(ky * Float64(sin(th) / hypot(ky, sin(kx))));
	else
		tmp = sin(th);
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if (sin(ky) <= -0.02)
		tmp = sin(ky) * (sin(th) / sin(kx));
	elseif (sin(ky) <= 0.02)
		tmp = ky * (sin(th) / hypot(ky, sin(kx)));
	else
		tmp = sin(th);
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[N[Sin[ky], $MachinePrecision], -0.02], N[(N[Sin[ky], $MachinePrecision] * N[(N[Sin[th], $MachinePrecision] / N[Sin[kx], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Sin[ky], $MachinePrecision], 0.02], N[(ky * N[(N[Sin[th], $MachinePrecision] / N[Sqrt[ky ^ 2 + N[Sin[kx], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Sin[th], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (sin.f64 ky) < -0.0200000000000000004

   1. Initial program 99.7%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Step-by-step derivation

      1. associate-*l/ N/A

         \[\leadsto \frac{\sin ky \cdot \sin th}{\color{blue}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}} \]

      2. associate-/l* N/A

         \[\leadsto \sin ky \cdot \color{blue}{\frac{\sin th}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\sin ky, \color{blue}{\left(\frac{\sin th}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}\right)}\right) \]

      4. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \left(\frac{\color{blue}{\sin th}}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}\right)\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\sin th, \color{blue}{\left(\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}\right)}\right)\right) \]

      6. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\mathsf{sin.f64}\left(th\right), \left(\sqrt{\color{blue}{{\sin kx}^{2} + {\sin ky}^{2}}}\right)\right)\right) \]

      7. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\mathsf{sin.f64}\left(th\right), \left(\sqrt{{\sin ky}^{2} + {\sin kx}^{2}}\right)\right)\right) \]

      8. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\mathsf{sin.f64}\left(th\right), \left(\sqrt{\sin ky \cdot \sin ky + {\sin kx}^{2}}\right)\right)\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\mathsf{sin.f64}\left(th\right), \left(\sqrt{\sin ky \cdot \sin ky + \sin kx \cdot \sin kx}\right)\right)\right) \]

      10. hypot-define N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\mathsf{sin.f64}\left(th\right), \left(\mathsf{hypot}\left(\sin ky, \color{blue}{\sin kx}\right)\right)\right)\right) \]

      11. hypot-lowering-hypot.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\mathsf{sin.f64}\left(th\right), \mathsf{hypot.f64}\left(\sin ky, \color{blue}{\sin kx}\right)\right)\right) \]

      12. sin-lowering-sin.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\mathsf{sin.f64}\left(th\right), \mathsf{hypot.f64}\left(\mathsf{sin.f64}\left(ky\right), \sin \color{blue}{kx}\right)\right)\right) \]

      13. sin-lowering-sin.f64 99.5%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\mathsf{sin.f64}\left(th\right), \mathsf{hypot.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{sin.f64}\left(kx\right)\right)\right)\right) \]

   3. Simplified 99.5%

      \[\leadsto \color{blue}{\sin ky \cdot \frac{\sin th}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in ky around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \color{blue}{\left(\frac{\sin th}{\sin kx}\right)}\right) \]
   6. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\sin th, \color{blue}{\sin kx}\right)\right) \]

      2. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\mathsf{sin.f64}\left(th\right), \sin \color{blue}{kx}\right)\right) \]

      3. sin-lowering-sin.f64 6.5%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\mathsf{sin.f64}\left(th\right), \mathsf{sin.f64}\left(kx\right)\right)\right) \]

   7. Simplified 6.5%

      \[\leadsto \sin ky \cdot \color{blue}{\frac{\sin th}{\sin kx}} \]

   if -0.0200000000000000004 < (sin.f64 ky) < 0.0200000000000000004

   1. Initial program 87.3%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Step-by-step derivation

      1. associate-*l/ N/A

         \[\leadsto \frac{\sin ky \cdot \sin th}{\color{blue}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}} \]

      2. associate-/l* N/A

         \[\leadsto \sin ky \cdot \color{blue}{\frac{\sin th}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\sin ky, \color{blue}{\left(\frac{\sin th}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}\right)}\right) \]

      4. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \left(\frac{\color{blue}{\sin th}}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}\right)\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\sin th, \color{blue}{\left(\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}\right)}\right)\right) \]

      6. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\mathsf{sin.f64}\left(th\right), \left(\sqrt{\color{blue}{{\sin kx}^{2} + {\sin ky}^{2}}}\right)\right)\right) \]

      7. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\mathsf{sin.f64}\left(th\right), \left(\sqrt{{\sin ky}^{2} + {\sin kx}^{2}}\right)\right)\right) \]

      8. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\mathsf{sin.f64}\left(th\right), \left(\sqrt{\sin ky \cdot \sin ky + {\sin kx}^{2}}\right)\right)\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\mathsf{sin.f64}\left(th\right), \left(\sqrt{\sin ky \cdot \sin ky + \sin kx \cdot \sin kx}\right)\right)\right) \]

      10. hypot-define N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\mathsf{sin.f64}\left(th\right), \left(\mathsf{hypot}\left(\sin ky, \color{blue}{\sin kx}\right)\right)\right)\right) \]

      11. hypot-lowering-hypot.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\mathsf{sin.f64}\left(th\right), \mathsf{hypot.f64}\left(\sin ky, \color{blue}{\sin kx}\right)\right)\right) \]

      12. sin-lowering-sin.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\mathsf{sin.f64}\left(th\right), \mathsf{hypot.f64}\left(\mathsf{sin.f64}\left(ky\right), \sin \color{blue}{kx}\right)\right)\right) \]

      13. sin-lowering-sin.f64 99.6%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\mathsf{sin.f64}\left(th\right), \mathsf{hypot.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{sin.f64}\left(kx\right)\right)\right)\right) \]

   3. Simplified 99.6%

      \[\leadsto \color{blue}{\sin ky \cdot \frac{\sin th}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in ky around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\mathsf{sin.f64}\left(th\right), \mathsf{hypot.f64}\left(\color{blue}{ky}, \mathsf{sin.f64}\left(kx\right)\right)\right)\right) \]
   6. Step-by-step derivation

   7. Simplified 98.9%

      \[\leadsto \sin ky \cdot \frac{\sin th}{\mathsf{hypot}\left(\color{blue}{ky}, \sin kx\right)} \]

   8. Taylor expanded in ky around 0

      \[\leadsto \mathsf{*.f64}\left(\color{blue}{ky}, \mathsf{/.f64}\left(\mathsf{sin.f64}\left(th\right), \mathsf{hypot.f64}\left(ky, \mathsf{sin.f64}\left(kx\right)\right)\right)\right) \]
   9. Step-by-step derivation

   10. Simplified 98.9%

       \[\leadsto \color{blue}{ky} \cdot \frac{\sin th}{\mathsf{hypot}\left(ky, \sin kx\right)} \]

   if 0.0200000000000000004 < (sin.f64 ky)

   1. Initial program 99.6%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Step-by-step derivation

      1. associate-*l/ N/A

         \[\leadsto \frac{\sin ky \cdot \sin th}{\color{blue}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}} \]

      2. associate-/l* N/A

         \[\leadsto \sin ky \cdot \color{blue}{\frac{\sin th}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\sin ky, \color{blue}{\left(\frac{\sin th}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}\right)}\right) \]

      4. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \left(\frac{\color{blue}{\sin th}}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}\right)\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\sin th, \color{blue}{\left(\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}\right)}\right)\right) \]

      6. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\mathsf{sin.f64}\left(th\right), \left(\sqrt{\color{blue}{{\sin kx}^{2} + {\sin ky}^{2}}}\right)\right)\right) \]

      7. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\mathsf{sin.f64}\left(th\right), \left(\sqrt{{\sin ky}^{2} + {\sin kx}^{2}}\right)\right)\right) \]

      8. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\mathsf{sin.f64}\left(th\right), \left(\sqrt{\sin ky \cdot \sin ky + {\sin kx}^{2}}\right)\right)\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\mathsf{sin.f64}\left(th\right), \left(\sqrt{\sin ky \cdot \sin ky + \sin kx \cdot \sin kx}\right)\right)\right) \]

      10. hypot-define N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\mathsf{sin.f64}\left(th\right), \left(\mathsf{hypot}\left(\sin ky, \color{blue}{\sin kx}\right)\right)\right)\right) \]

      11. hypot-lowering-hypot.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\mathsf{sin.f64}\left(th\right), \mathsf{hypot.f64}\left(\sin ky, \color{blue}{\sin kx}\right)\right)\right) \]

      12. sin-lowering-sin.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\mathsf{sin.f64}\left(th\right), \mathsf{hypot.f64}\left(\mathsf{sin.f64}\left(ky\right), \sin \color{blue}{kx}\right)\right)\right) \]

      13. sin-lowering-sin.f64 99.6%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\mathsf{sin.f64}\left(th\right), \mathsf{hypot.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{sin.f64}\left(kx\right)\right)\right)\right) \]

   3. Simplified 99.6%

      \[\leadsto \color{blue}{\sin ky \cdot \frac{\sin th}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in kx around 0

      \[\leadsto \color{blue}{\sin th} \]
   6. Step-by-step derivation

      1. sin-lowering-sin.f64 61.0%

         \[\leadsto \mathsf{sin.f64}\left(th\right) \]

   7. Simplified 61.0%

      \[\leadsto \color{blue}{\sin th} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 6: 99.6% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \sin ky \cdot \frac{\sin th}{\mathsf{hypot}\left(\sin ky, \sin kx\right)} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (* (sin ky) (/ (sin th) (hypot (sin ky) (sin kx)))))

C

double code(double kx, double ky, double th) {
	return sin(ky) * (sin(th) / hypot(sin(ky), sin(kx)));
}

Java

public static double code(double kx, double ky, double th) {
	return Math.sin(ky) * (Math.sin(th) / Math.hypot(Math.sin(ky), Math.sin(kx)));
}

Python

def code(kx, ky, th):
	return math.sin(ky) * (math.sin(th) / math.hypot(math.sin(ky), math.sin(kx)))

Julia

function code(kx, ky, th)
	return Float64(sin(ky) * Float64(sin(th) / hypot(sin(ky), sin(kx))))
end

MATLAB

function tmp = code(kx, ky, th)
	tmp = sin(ky) * (sin(th) / hypot(sin(ky), sin(kx)));
end

Wolfram

code[kx_, ky_, th_] := N[(N[Sin[ky], $MachinePrecision] * N[(N[Sin[th], $MachinePrecision] / N[Sqrt[N[Sin[ky], $MachinePrecision] ^ 2 + N[Sin[kx], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 93.7%

   \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
2. Step-by-step derivation

   1. associate-*l/ N/A

      \[\leadsto \frac{\sin ky \cdot \sin th}{\color{blue}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}} \]

   2. associate-/l* N/A

      \[\leadsto \sin ky \cdot \color{blue}{\frac{\sin th}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}} \]

   3. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\sin ky, \color{blue}{\left(\frac{\sin th}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}\right)}\right) \]

   4. sin-lowering-sin.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \left(\frac{\color{blue}{\sin th}}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}\right)\right) \]

   5. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\sin th, \color{blue}{\left(\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}\right)}\right)\right) \]

   6. sin-lowering-sin.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\mathsf{sin.f64}\left(th\right), \left(\sqrt{\color{blue}{{\sin kx}^{2} + {\sin ky}^{2}}}\right)\right)\right) \]

   7. +-commutative N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\mathsf{sin.f64}\left(th\right), \left(\sqrt{{\sin ky}^{2} + {\sin kx}^{2}}\right)\right)\right) \]

   8. unpow2 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\mathsf{sin.f64}\left(th\right), \left(\sqrt{\sin ky \cdot \sin ky + {\sin kx}^{2}}\right)\right)\right) \]

   9. unpow2 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\mathsf{sin.f64}\left(th\right), \left(\sqrt{\sin ky \cdot \sin ky + \sin kx \cdot \sin kx}\right)\right)\right) \]

   10. hypot-define N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\mathsf{sin.f64}\left(th\right), \left(\mathsf{hypot}\left(\sin ky, \color{blue}{\sin kx}\right)\right)\right)\right) \]

   11. hypot-lowering-hypot.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\mathsf{sin.f64}\left(th\right), \mathsf{hypot.f64}\left(\sin ky, \color{blue}{\sin kx}\right)\right)\right) \]

   12. sin-lowering-sin.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\mathsf{sin.f64}\left(th\right), \mathsf{hypot.f64}\left(\mathsf{sin.f64}\left(ky\right), \sin \color{blue}{kx}\right)\right)\right) \]

   13. sin-lowering-sin.f64 99.6%

       \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\mathsf{sin.f64}\left(th\right), \mathsf{hypot.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{sin.f64}\left(kx\right)\right)\right)\right) \]

3. Simplified 99.6%

   \[\leadsto \color{blue}{\sin ky \cdot \frac{\sin th}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}} \]
4. Add Preprocessing
5. Add Preprocessing

Alternative 7: 67.0% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;th \leq 0.095:\\ \;\;\;\;\sin ky \cdot \frac{th \cdot \left(1 + th \cdot \left(th \cdot \left(-0.16666666666666666 + 0.008333333333333333 \cdot \left(th \cdot th\right)\right)\right)\right)}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}\\ \mathbf{else}:\\ \;\;\;\;ky \cdot \frac{\sin th}{\mathsf{hypot}\left(ky, \sin kx\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= th 0.095)
   (*
    (sin ky)
    (/
     (*
      th
      (+
       1.0
       (*
        th
        (* th (+ -0.16666666666666666 (* 0.008333333333333333 (* th th)))))))
     (hypot (sin ky) (sin kx))))
   (* ky (/ (sin th) (hypot ky (sin kx))))))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (th <= 0.095) {
		tmp = sin(ky) * ((th * (1.0 + (th * (th * (-0.16666666666666666 + (0.008333333333333333 * (th * th))))))) / hypot(sin(ky), sin(kx)));
	} else {
		tmp = ky * (sin(th) / hypot(ky, sin(kx)));
	}
	return tmp;
}

Java

public static double code(double kx, double ky, double th) {
	double tmp;
	if (th <= 0.095) {
		tmp = Math.sin(ky) * ((th * (1.0 + (th * (th * (-0.16666666666666666 + (0.008333333333333333 * (th * th))))))) / Math.hypot(Math.sin(ky), Math.sin(kx)));
	} else {
		tmp = ky * (Math.sin(th) / Math.hypot(ky, Math.sin(kx)));
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if th <= 0.095:
		tmp = math.sin(ky) * ((th * (1.0 + (th * (th * (-0.16666666666666666 + (0.008333333333333333 * (th * th))))))) / math.hypot(math.sin(ky), math.sin(kx)))
	else:
		tmp = ky * (math.sin(th) / math.hypot(ky, math.sin(kx)))
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (th <= 0.095)
		tmp = Float64(sin(ky) * Float64(Float64(th * Float64(1.0 + Float64(th * Float64(th * Float64(-0.16666666666666666 + Float64(0.008333333333333333 * Float64(th * th))))))) / hypot(sin(ky), sin(kx))));
	else
		tmp = Float64(ky * Float64(sin(th) / hypot(ky, sin(kx))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if (th <= 0.095)
		tmp = sin(ky) * ((th * (1.0 + (th * (th * (-0.16666666666666666 + (0.008333333333333333 * (th * th))))))) / hypot(sin(ky), sin(kx)));
	else
		tmp = ky * (sin(th) / hypot(ky, sin(kx)));
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[th, 0.095], N[(N[Sin[ky], $MachinePrecision] * N[(N[(th * N[(1.0 + N[(th * N[(th * N[(-0.16666666666666666 + N[(0.008333333333333333 * N[(th * th), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Sqrt[N[Sin[ky], $MachinePrecision] ^ 2 + N[Sin[kx], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(ky * N[(N[Sin[th], $MachinePrecision] / N[Sqrt[ky ^ 2 + N[Sin[kx], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if th < 0.095000000000000001

   1. Initial program 92.1%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Step-by-step derivation

      1. associate-*l/ N/A

         \[\leadsto \frac{\sin ky \cdot \sin th}{\color{blue}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}} \]

      2. associate-/l* N/A

         \[\leadsto \sin ky \cdot \color{blue}{\frac{\sin th}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\sin ky, \color{blue}{\left(\frac{\sin th}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}\right)}\right) \]

      4. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \left(\frac{\color{blue}{\sin th}}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}\right)\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\sin th, \color{blue}{\left(\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}\right)}\right)\right) \]

      6. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\mathsf{sin.f64}\left(th\right), \left(\sqrt{\color{blue}{{\sin kx}^{2} + {\sin ky}^{2}}}\right)\right)\right) \]

      7. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\mathsf{sin.f64}\left(th\right), \left(\sqrt{{\sin ky}^{2} + {\sin kx}^{2}}\right)\right)\right) \]

      8. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\mathsf{sin.f64}\left(th\right), \left(\sqrt{\sin ky \cdot \sin ky + {\sin kx}^{2}}\right)\right)\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\mathsf{sin.f64}\left(th\right), \left(\sqrt{\sin ky \cdot \sin ky + \sin kx \cdot \sin kx}\right)\right)\right) \]

      10. hypot-define N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\mathsf{sin.f64}\left(th\right), \left(\mathsf{hypot}\left(\sin ky, \color{blue}{\sin kx}\right)\right)\right)\right) \]

      11. hypot-lowering-hypot.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\mathsf{sin.f64}\left(th\right), \mathsf{hypot.f64}\left(\sin ky, \color{blue}{\sin kx}\right)\right)\right) \]

      12. sin-lowering-sin.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\mathsf{sin.f64}\left(th\right), \mathsf{hypot.f64}\left(\mathsf{sin.f64}\left(ky\right), \sin \color{blue}{kx}\right)\right)\right) \]

      13. sin-lowering-sin.f64 99.6%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\mathsf{sin.f64}\left(th\right), \mathsf{hypot.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{sin.f64}\left(kx\right)\right)\right)\right) \]

   3. Simplified 99.6%

      \[\leadsto \color{blue}{\sin ky \cdot \frac{\sin th}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in th around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\color{blue}{\left(th \cdot \left(1 + {th}^{2} \cdot \left(\frac{1}{120} \cdot {th}^{2} - \frac{1}{6}\right)\right)\right)}, \mathsf{hypot.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{sin.f64}\left(kx\right)\right)\right)\right) \]
   6. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(th, \left(1 + {th}^{2} \cdot \left(\frac{1}{120} \cdot {th}^{2} - \frac{1}{6}\right)\right)\right), \mathsf{hypot.f64}\left(\color{blue}{\mathsf{sin.f64}\left(ky\right)}, \mathsf{sin.f64}\left(kx\right)\right)\right)\right) \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(th, \mathsf{+.f64}\left(1, \left({th}^{2} \cdot \left(\frac{1}{120} \cdot {th}^{2} - \frac{1}{6}\right)\right)\right)\right), \mathsf{hypot.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{sin.f64}\left(kx\right)\right)\right)\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(th, \mathsf{+.f64}\left(1, \left(\left(th \cdot th\right) \cdot \left(\frac{1}{120} \cdot {th}^{2} - \frac{1}{6}\right)\right)\right)\right), \mathsf{hypot.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{sin.f64}\left(kx\right)\right)\right)\right) \]

      4. associate-*l* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(th, \mathsf{+.f64}\left(1, \left(th \cdot \left(th \cdot \left(\frac{1}{120} \cdot {th}^{2} - \frac{1}{6}\right)\right)\right)\right)\right), \mathsf{hypot.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{sin.f64}\left(kx\right)\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(th, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(th, \left(th \cdot \left(\frac{1}{120} \cdot {th}^{2} - \frac{1}{6}\right)\right)\right)\right)\right), \mathsf{hypot.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{sin.f64}\left(kx\right)\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(th, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(th, \mathsf{*.f64}\left(th, \left(\frac{1}{120} \cdot {th}^{2} - \frac{1}{6}\right)\right)\right)\right)\right), \mathsf{hypot.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{sin.f64}\left(kx\right)\right)\right)\right) \]

      7. sub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(th, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(th, \mathsf{*.f64}\left(th, \left(\frac{1}{120} \cdot {th}^{2} + \left(\mathsf{neg}\left(\frac{1}{6}\right)\right)\right)\right)\right)\right)\right), \mathsf{hypot.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{sin.f64}\left(kx\right)\right)\right)\right) \]

      8. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(th, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(th, \mathsf{*.f64}\left(th, \left(\frac{1}{120} \cdot {th}^{2} + \frac{-1}{6}\right)\right)\right)\right)\right), \mathsf{hypot.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{sin.f64}\left(kx\right)\right)\right)\right) \]

      9. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(th, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(th, \mathsf{*.f64}\left(th, \left(\frac{-1}{6} + \frac{1}{120} \cdot {th}^{2}\right)\right)\right)\right)\right), \mathsf{hypot.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{sin.f64}\left(kx\right)\right)\right)\right) \]

      10. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(th, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(th, \mathsf{*.f64}\left(th, \mathsf{+.f64}\left(\frac{-1}{6}, \left(\frac{1}{120} \cdot {th}^{2}\right)\right)\right)\right)\right)\right), \mathsf{hypot.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{sin.f64}\left(kx\right)\right)\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(th, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(th, \mathsf{*.f64}\left(th, \mathsf{+.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(\frac{1}{120}, \left({th}^{2}\right)\right)\right)\right)\right)\right)\right), \mathsf{hypot.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{sin.f64}\left(kx\right)\right)\right)\right) \]

      12. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(th, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(th, \mathsf{*.f64}\left(th, \mathsf{+.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(\frac{1}{120}, \left(th \cdot th\right)\right)\right)\right)\right)\right)\right), \mathsf{hypot.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{sin.f64}\left(kx\right)\right)\right)\right) \]

      13. *-lowering-*.f64 63.0%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(th, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(th, \mathsf{*.f64}\left(th, \mathsf{+.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(\frac{1}{120}, \mathsf{*.f64}\left(th, th\right)\right)\right)\right)\right)\right)\right), \mathsf{hypot.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{sin.f64}\left(kx\right)\right)\right)\right) \]

   7. Simplified 63.0%

      \[\leadsto \sin ky \cdot \frac{\color{blue}{th \cdot \left(1 + th \cdot \left(th \cdot \left(-0.16666666666666666 + 0.008333333333333333 \cdot \left(th \cdot th\right)\right)\right)\right)}}{\mathsf{hypot}\left(\sin ky, \sin kx\right)} \]

   if 0.095000000000000001 < th

   1. Initial program 98.2%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Step-by-step derivation

      1. associate-*l/ N/A

         \[\leadsto \frac{\sin ky \cdot \sin th}{\color{blue}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}} \]

      2. associate-/l* N/A

         \[\leadsto \sin ky \cdot \color{blue}{\frac{\sin th}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\sin ky, \color{blue}{\left(\frac{\sin th}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}\right)}\right) \]

      4. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \left(\frac{\color{blue}{\sin th}}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}\right)\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\sin th, \color{blue}{\left(\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}\right)}\right)\right) \]

      6. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\mathsf{sin.f64}\left(th\right), \left(\sqrt{\color{blue}{{\sin kx}^{2} + {\sin ky}^{2}}}\right)\right)\right) \]

      7. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\mathsf{sin.f64}\left(th\right), \left(\sqrt{{\sin ky}^{2} + {\sin kx}^{2}}\right)\right)\right) \]

      8. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\mathsf{sin.f64}\left(th\right), \left(\sqrt{\sin ky \cdot \sin ky + {\sin kx}^{2}}\right)\right)\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\mathsf{sin.f64}\left(th\right), \left(\sqrt{\sin ky \cdot \sin ky + \sin kx \cdot \sin kx}\right)\right)\right) \]

      10. hypot-define N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\mathsf{sin.f64}\left(th\right), \left(\mathsf{hypot}\left(\sin ky, \color{blue}{\sin kx}\right)\right)\right)\right) \]

      11. hypot-lowering-hypot.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\mathsf{sin.f64}\left(th\right), \mathsf{hypot.f64}\left(\sin ky, \color{blue}{\sin kx}\right)\right)\right) \]

      12. sin-lowering-sin.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\mathsf{sin.f64}\left(th\right), \mathsf{hypot.f64}\left(\mathsf{sin.f64}\left(ky\right), \sin \color{blue}{kx}\right)\right)\right) \]

      13. sin-lowering-sin.f64 99.6%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\mathsf{sin.f64}\left(th\right), \mathsf{hypot.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{sin.f64}\left(kx\right)\right)\right)\right) \]

   3. Simplified 99.6%

      \[\leadsto \color{blue}{\sin ky \cdot \frac{\sin th}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in ky around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\mathsf{sin.f64}\left(th\right), \mathsf{hypot.f64}\left(\color{blue}{ky}, \mathsf{sin.f64}\left(kx\right)\right)\right)\right) \]
   6. Step-by-step derivation

   7. Simplified 48.1%

      \[\leadsto \sin ky \cdot \frac{\sin th}{\mathsf{hypot}\left(\color{blue}{ky}, \sin kx\right)} \]

   8. Taylor expanded in ky around 0

      \[\leadsto \mathsf{*.f64}\left(\color{blue}{ky}, \mathsf{/.f64}\left(\mathsf{sin.f64}\left(th\right), \mathsf{hypot.f64}\left(ky, \mathsf{sin.f64}\left(kx\right)\right)\right)\right) \]
   9. Step-by-step derivation

   10. Simplified 61.7%

       \[\leadsto \color{blue}{ky} \cdot \frac{\sin th}{\mathsf{hypot}\left(ky, \sin kx\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 8: 67.0% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;th \leq 0.024:\\ \;\;\;\;\sin ky \cdot \frac{th \cdot \left(1 + th \cdot \left(th \cdot -0.16666666666666666\right)\right)}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}\\ \mathbf{else}:\\ \;\;\;\;ky \cdot \frac{\sin th}{\mathsf{hypot}\left(ky, \sin kx\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= th 0.024)
   (*
    (sin ky)
    (/
     (* th (+ 1.0 (* th (* th -0.16666666666666666))))
     (hypot (sin ky) (sin kx))))
   (* ky (/ (sin th) (hypot ky (sin kx))))))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (th <= 0.024) {
		tmp = sin(ky) * ((th * (1.0 + (th * (th * -0.16666666666666666)))) / hypot(sin(ky), sin(kx)));
	} else {
		tmp = ky * (sin(th) / hypot(ky, sin(kx)));
	}
	return tmp;
}

Java

public static double code(double kx, double ky, double th) {
	double tmp;
	if (th <= 0.024) {
		tmp = Math.sin(ky) * ((th * (1.0 + (th * (th * -0.16666666666666666)))) / Math.hypot(Math.sin(ky), Math.sin(kx)));
	} else {
		tmp = ky * (Math.sin(th) / Math.hypot(ky, Math.sin(kx)));
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if th <= 0.024:
		tmp = math.sin(ky) * ((th * (1.0 + (th * (th * -0.16666666666666666)))) / math.hypot(math.sin(ky), math.sin(kx)))
	else:
		tmp = ky * (math.sin(th) / math.hypot(ky, math.sin(kx)))
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (th <= 0.024)
		tmp = Float64(sin(ky) * Float64(Float64(th * Float64(1.0 + Float64(th * Float64(th * -0.16666666666666666)))) / hypot(sin(ky), sin(kx))));
	else
		tmp = Float64(ky * Float64(sin(th) / hypot(ky, sin(kx))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if (th <= 0.024)
		tmp = sin(ky) * ((th * (1.0 + (th * (th * -0.16666666666666666)))) / hypot(sin(ky), sin(kx)));
	else
		tmp = ky * (sin(th) / hypot(ky, sin(kx)));
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[th, 0.024], N[(N[Sin[ky], $MachinePrecision] * N[(N[(th * N[(1.0 + N[(th * N[(th * -0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Sqrt[N[Sin[ky], $MachinePrecision] ^ 2 + N[Sin[kx], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(ky * N[(N[Sin[th], $MachinePrecision] / N[Sqrt[ky ^ 2 + N[Sin[kx], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if th < 0.024

   1. Initial program 92.1%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Step-by-step derivation

      1. associate-*l/ N/A

         \[\leadsto \frac{\sin ky \cdot \sin th}{\color{blue}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}} \]

      2. associate-/l* N/A

         \[\leadsto \sin ky \cdot \color{blue}{\frac{\sin th}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\sin ky, \color{blue}{\left(\frac{\sin th}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}\right)}\right) \]

      4. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \left(\frac{\color{blue}{\sin th}}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}\right)\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\sin th, \color{blue}{\left(\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}\right)}\right)\right) \]

      6. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\mathsf{sin.f64}\left(th\right), \left(\sqrt{\color{blue}{{\sin kx}^{2} + {\sin ky}^{2}}}\right)\right)\right) \]

      7. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\mathsf{sin.f64}\left(th\right), \left(\sqrt{{\sin ky}^{2} + {\sin kx}^{2}}\right)\right)\right) \]

      8. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\mathsf{sin.f64}\left(th\right), \left(\sqrt{\sin ky \cdot \sin ky + {\sin kx}^{2}}\right)\right)\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\mathsf{sin.f64}\left(th\right), \left(\sqrt{\sin ky \cdot \sin ky + \sin kx \cdot \sin kx}\right)\right)\right) \]

      10. hypot-define N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\mathsf{sin.f64}\left(th\right), \left(\mathsf{hypot}\left(\sin ky, \color{blue}{\sin kx}\right)\right)\right)\right) \]

      11. hypot-lowering-hypot.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\mathsf{sin.f64}\left(th\right), \mathsf{hypot.f64}\left(\sin ky, \color{blue}{\sin kx}\right)\right)\right) \]

      12. sin-lowering-sin.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\mathsf{sin.f64}\left(th\right), \mathsf{hypot.f64}\left(\mathsf{sin.f64}\left(ky\right), \sin \color{blue}{kx}\right)\right)\right) \]

      13. sin-lowering-sin.f64 99.6%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\mathsf{sin.f64}\left(th\right), \mathsf{hypot.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{sin.f64}\left(kx\right)\right)\right)\right) \]

   3. Simplified 99.6%

      \[\leadsto \color{blue}{\sin ky \cdot \frac{\sin th}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in th around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\color{blue}{\left(th \cdot \left(1 + \frac{-1}{6} \cdot {th}^{2}\right)\right)}, \mathsf{hypot.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{sin.f64}\left(kx\right)\right)\right)\right) \]
   6. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(th, \left(1 + \frac{-1}{6} \cdot {th}^{2}\right)\right), \mathsf{hypot.f64}\left(\color{blue}{\mathsf{sin.f64}\left(ky\right)}, \mathsf{sin.f64}\left(kx\right)\right)\right)\right) \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(th, \mathsf{+.f64}\left(1, \left(\frac{-1}{6} \cdot {th}^{2}\right)\right)\right), \mathsf{hypot.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{sin.f64}\left(kx\right)\right)\right)\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(th, \mathsf{+.f64}\left(1, \left(\frac{-1}{6} \cdot \left(th \cdot th\right)\right)\right)\right), \mathsf{hypot.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{sin.f64}\left(kx\right)\right)\right)\right) \]

      4. associate-*r* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(th, \mathsf{+.f64}\left(1, \left(\left(\frac{-1}{6} \cdot th\right) \cdot th\right)\right)\right), \mathsf{hypot.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{sin.f64}\left(kx\right)\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(th, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(\frac{-1}{6} \cdot th\right), th\right)\right)\right), \mathsf{hypot.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{sin.f64}\left(kx\right)\right)\right)\right) \]

      6. *-lowering-*.f64 62.9%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(th, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-1}{6}, th\right), th\right)\right)\right), \mathsf{hypot.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{sin.f64}\left(kx\right)\right)\right)\right) \]

   7. Simplified 62.9%

      \[\leadsto \sin ky \cdot \frac{\color{blue}{th \cdot \left(1 + \left(-0.16666666666666666 \cdot th\right) \cdot th\right)}}{\mathsf{hypot}\left(\sin ky, \sin kx\right)} \]

   if 0.024 < th

   1. Initial program 98.2%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Step-by-step derivation

      1. associate-*l/ N/A

         \[\leadsto \frac{\sin ky \cdot \sin th}{\color{blue}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}} \]

      2. associate-/l* N/A

         \[\leadsto \sin ky \cdot \color{blue}{\frac{\sin th}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\sin ky, \color{blue}{\left(\frac{\sin th}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}\right)}\right) \]

      4. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \left(\frac{\color{blue}{\sin th}}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}\right)\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\sin th, \color{blue}{\left(\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}\right)}\right)\right) \]

      6. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\mathsf{sin.f64}\left(th\right), \left(\sqrt{\color{blue}{{\sin kx}^{2} + {\sin ky}^{2}}}\right)\right)\right) \]

      7. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\mathsf{sin.f64}\left(th\right), \left(\sqrt{{\sin ky}^{2} + {\sin kx}^{2}}\right)\right)\right) \]

      8. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\mathsf{sin.f64}\left(th\right), \left(\sqrt{\sin ky \cdot \sin ky + {\sin kx}^{2}}\right)\right)\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\mathsf{sin.f64}\left(th\right), \left(\sqrt{\sin ky \cdot \sin ky + \sin kx \cdot \sin kx}\right)\right)\right) \]

      10. hypot-define N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\mathsf{sin.f64}\left(th\right), \left(\mathsf{hypot}\left(\sin ky, \color{blue}{\sin kx}\right)\right)\right)\right) \]

      11. hypot-lowering-hypot.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\mathsf{sin.f64}\left(th\right), \mathsf{hypot.f64}\left(\sin ky, \color{blue}{\sin kx}\right)\right)\right) \]

      12. sin-lowering-sin.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\mathsf{sin.f64}\left(th\right), \mathsf{hypot.f64}\left(\mathsf{sin.f64}\left(ky\right), \sin \color{blue}{kx}\right)\right)\right) \]

      13. sin-lowering-sin.f64 99.6%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\mathsf{sin.f64}\left(th\right), \mathsf{hypot.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{sin.f64}\left(kx\right)\right)\right)\right) \]

   3. Simplified 99.6%

      \[\leadsto \color{blue}{\sin ky \cdot \frac{\sin th}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in ky around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\mathsf{sin.f64}\left(th\right), \mathsf{hypot.f64}\left(\color{blue}{ky}, \mathsf{sin.f64}\left(kx\right)\right)\right)\right) \]
   6. Step-by-step derivation

   7. Simplified 48.1%

      \[\leadsto \sin ky \cdot \frac{\sin th}{\mathsf{hypot}\left(\color{blue}{ky}, \sin kx\right)} \]

   8. Taylor expanded in ky around 0

      \[\leadsto \mathsf{*.f64}\left(\color{blue}{ky}, \mathsf{/.f64}\left(\mathsf{sin.f64}\left(th\right), \mathsf{hypot.f64}\left(ky, \mathsf{sin.f64}\left(kx\right)\right)\right)\right) \]
   9. Step-by-step derivation

   10. Simplified 61.7%

       \[\leadsto \color{blue}{ky} \cdot \frac{\sin th}{\mathsf{hypot}\left(ky, \sin kx\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 62.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;th \leq 0.024:\\ \;\;\;\;\sin ky \cdot \frac{th \cdot \left(1 + th \cdot \left(th \cdot -0.16666666666666666\right)\right)}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}\\ \mathbf{else}:\\ \;\;\;\;ky \cdot \frac{\sin th}{\mathsf{hypot}\left(ky, \sin kx\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 67.0% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;th \leq 0.0025:\\ \;\;\;\;\frac{\sin ky}{\mathsf{hypot}\left(\sin kx, \sin ky\right)} \cdot th\\ \mathbf{else}:\\ \;\;\;\;ky \cdot \frac{\sin th}{\mathsf{hypot}\left(ky, \sin kx\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= th 0.0025)
   (* (/ (sin ky) (hypot (sin kx) (sin ky))) th)
   (* ky (/ (sin th) (hypot ky (sin kx))))))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (th <= 0.0025) {
		tmp = (sin(ky) / hypot(sin(kx), sin(ky))) * th;
	} else {
		tmp = ky * (sin(th) / hypot(ky, sin(kx)));
	}
	return tmp;
}

Java

public static double code(double kx, double ky, double th) {
	double tmp;
	if (th <= 0.0025) {
		tmp = (Math.sin(ky) / Math.hypot(Math.sin(kx), Math.sin(ky))) * th;
	} else {
		tmp = ky * (Math.sin(th) / Math.hypot(ky, Math.sin(kx)));
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if th <= 0.0025:
		tmp = (math.sin(ky) / math.hypot(math.sin(kx), math.sin(ky))) * th
	else:
		tmp = ky * (math.sin(th) / math.hypot(ky, math.sin(kx)))
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (th <= 0.0025)
		tmp = Float64(Float64(sin(ky) / hypot(sin(kx), sin(ky))) * th);
	else
		tmp = Float64(ky * Float64(sin(th) / hypot(ky, sin(kx))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if (th <= 0.0025)
		tmp = (sin(ky) / hypot(sin(kx), sin(ky))) * th;
	else
		tmp = ky * (sin(th) / hypot(ky, sin(kx)));
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[th, 0.0025], N[(N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[Sin[kx], $MachinePrecision] ^ 2 + N[Sin[ky], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision] * th), $MachinePrecision], N[(ky * N[(N[Sin[th], $MachinePrecision] / N[Sqrt[ky ^ 2 + N[Sin[kx], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if th < 0.00250000000000000005

   1. Initial program 92.1%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing

   3. Taylor expanded in kx around inf

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sin.f64}\left(ky\right), \color{blue}{\left(\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}\right)}\right), \mathsf{sin.f64}\left(th\right)\right) \]
   4. Step-by-step derivation

      1. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sin.f64}\left(ky\right), \left(\sqrt{\sin kx \cdot \sin kx + {\sin ky}^{2}}\right)\right), \mathsf{sin.f64}\left(th\right)\right) \]

      2. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sin.f64}\left(ky\right), \left(\sqrt{\sin kx \cdot \sin kx + \sin ky \cdot \sin ky}\right)\right), \mathsf{sin.f64}\left(th\right)\right) \]

      3. hypot-define N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sin.f64}\left(ky\right), \left(\mathsf{hypot}\left(\sin kx, \sin ky\right)\right)\right), \mathsf{sin.f64}\left(th\right)\right) \]

      4. hypot-lowering-hypot.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{hypot.f64}\left(\sin kx, \sin ky\right)\right), \mathsf{sin.f64}\left(th\right)\right) \]

      5. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{hypot.f64}\left(\mathsf{sin.f64}\left(kx\right), \sin ky\right)\right), \mathsf{sin.f64}\left(th\right)\right) \]

      6. sin-lowering-sin.f64 99.7%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{hypot.f64}\left(\mathsf{sin.f64}\left(kx\right), \mathsf{sin.f64}\left(ky\right)\right)\right), \mathsf{sin.f64}\left(th\right)\right) \]

   5. Simplified 99.7%

      \[\leadsto \frac{\sin ky}{\color{blue}{\mathsf{hypot}\left(\sin kx, \sin ky\right)}} \cdot \sin th \]

   6. Taylor expanded in th around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{hypot.f64}\left(\mathsf{sin.f64}\left(kx\right), \mathsf{sin.f64}\left(ky\right)\right)\right), \color{blue}{th}\right) \]
   7. Step-by-step derivation

   8. Simplified 63.0%

      \[\leadsto \frac{\sin ky}{\mathsf{hypot}\left(\sin kx, \sin ky\right)} \cdot \color{blue}{th} \]

   if 0.00250000000000000005 < th

   1. Initial program 98.2%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Step-by-step derivation

      1. associate-*l/ N/A

         \[\leadsto \frac{\sin ky \cdot \sin th}{\color{blue}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}} \]

      2. associate-/l* N/A

         \[\leadsto \sin ky \cdot \color{blue}{\frac{\sin th}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\sin ky, \color{blue}{\left(\frac{\sin th}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}\right)}\right) \]

      4. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \left(\frac{\color{blue}{\sin th}}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}\right)\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\sin th, \color{blue}{\left(\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}\right)}\right)\right) \]

      6. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\mathsf{sin.f64}\left(th\right), \left(\sqrt{\color{blue}{{\sin kx}^{2} + {\sin ky}^{2}}}\right)\right)\right) \]

      7. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\mathsf{sin.f64}\left(th\right), \left(\sqrt{{\sin ky}^{2} + {\sin kx}^{2}}\right)\right)\right) \]

      8. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\mathsf{sin.f64}\left(th\right), \left(\sqrt{\sin ky \cdot \sin ky + {\sin kx}^{2}}\right)\right)\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\mathsf{sin.f64}\left(th\right), \left(\sqrt{\sin ky \cdot \sin ky + \sin kx \cdot \sin kx}\right)\right)\right) \]

      10. hypot-define N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\mathsf{sin.f64}\left(th\right), \left(\mathsf{hypot}\left(\sin ky, \color{blue}{\sin kx}\right)\right)\right)\right) \]

      11. hypot-lowering-hypot.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\mathsf{sin.f64}\left(th\right), \mathsf{hypot.f64}\left(\sin ky, \color{blue}{\sin kx}\right)\right)\right) \]

      12. sin-lowering-sin.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\mathsf{sin.f64}\left(th\right), \mathsf{hypot.f64}\left(\mathsf{sin.f64}\left(ky\right), \sin \color{blue}{kx}\right)\right)\right) \]

      13. sin-lowering-sin.f64 99.6%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\mathsf{sin.f64}\left(th\right), \mathsf{hypot.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{sin.f64}\left(kx\right)\right)\right)\right) \]

   3. Simplified 99.6%

      \[\leadsto \color{blue}{\sin ky \cdot \frac{\sin th}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in ky around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\mathsf{sin.f64}\left(th\right), \mathsf{hypot.f64}\left(\color{blue}{ky}, \mathsf{sin.f64}\left(kx\right)\right)\right)\right) \]
   6. Step-by-step derivation

   7. Simplified 48.1%

      \[\leadsto \sin ky \cdot \frac{\sin th}{\mathsf{hypot}\left(\color{blue}{ky}, \sin kx\right)} \]

   8. Taylor expanded in ky around 0

      \[\leadsto \mathsf{*.f64}\left(\color{blue}{ky}, \mathsf{/.f64}\left(\mathsf{sin.f64}\left(th\right), \mathsf{hypot.f64}\left(ky, \mathsf{sin.f64}\left(kx\right)\right)\right)\right) \]
   9. Step-by-step derivation

   10. Simplified 61.7%

       \[\leadsto \color{blue}{ky} \cdot \frac{\sin th}{\mathsf{hypot}\left(ky, \sin kx\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 10: 67.0% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;th \leq 0.00225:\\ \;\;\;\;\sin ky \cdot \frac{th}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}\\ \mathbf{else}:\\ \;\;\;\;ky \cdot \frac{\sin th}{\mathsf{hypot}\left(ky, \sin kx\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= th 0.00225)
   (* (sin ky) (/ th (hypot (sin ky) (sin kx))))
   (* ky (/ (sin th) (hypot ky (sin kx))))))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (th <= 0.00225) {
		tmp = sin(ky) * (th / hypot(sin(ky), sin(kx)));
	} else {
		tmp = ky * (sin(th) / hypot(ky, sin(kx)));
	}
	return tmp;
}

Java

public static double code(double kx, double ky, double th) {
	double tmp;
	if (th <= 0.00225) {
		tmp = Math.sin(ky) * (th / Math.hypot(Math.sin(ky), Math.sin(kx)));
	} else {
		tmp = ky * (Math.sin(th) / Math.hypot(ky, Math.sin(kx)));
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if th <= 0.00225:
		tmp = math.sin(ky) * (th / math.hypot(math.sin(ky), math.sin(kx)))
	else:
		tmp = ky * (math.sin(th) / math.hypot(ky, math.sin(kx)))
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (th <= 0.00225)
		tmp = Float64(sin(ky) * Float64(th / hypot(sin(ky), sin(kx))));
	else
		tmp = Float64(ky * Float64(sin(th) / hypot(ky, sin(kx))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if (th <= 0.00225)
		tmp = sin(ky) * (th / hypot(sin(ky), sin(kx)));
	else
		tmp = ky * (sin(th) / hypot(ky, sin(kx)));
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[th, 0.00225], N[(N[Sin[ky], $MachinePrecision] * N[(th / N[Sqrt[N[Sin[ky], $MachinePrecision] ^ 2 + N[Sin[kx], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(ky * N[(N[Sin[th], $MachinePrecision] / N[Sqrt[ky ^ 2 + N[Sin[kx], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if th < 0.00224999999999999983

   1. Initial program 92.1%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Step-by-step derivation

      1. associate-*l/ N/A

         \[\leadsto \frac{\sin ky \cdot \sin th}{\color{blue}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}} \]

      2. associate-/l* N/A

         \[\leadsto \sin ky \cdot \color{blue}{\frac{\sin th}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\sin ky, \color{blue}{\left(\frac{\sin th}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}\right)}\right) \]

      4. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \left(\frac{\color{blue}{\sin th}}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}\right)\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\sin th, \color{blue}{\left(\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}\right)}\right)\right) \]

      6. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\mathsf{sin.f64}\left(th\right), \left(\sqrt{\color{blue}{{\sin kx}^{2} + {\sin ky}^{2}}}\right)\right)\right) \]

      7. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\mathsf{sin.f64}\left(th\right), \left(\sqrt{{\sin ky}^{2} + {\sin kx}^{2}}\right)\right)\right) \]

      8. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\mathsf{sin.f64}\left(th\right), \left(\sqrt{\sin ky \cdot \sin ky + {\sin kx}^{2}}\right)\right)\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\mathsf{sin.f64}\left(th\right), \left(\sqrt{\sin ky \cdot \sin ky + \sin kx \cdot \sin kx}\right)\right)\right) \]

      10. hypot-define N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\mathsf{sin.f64}\left(th\right), \left(\mathsf{hypot}\left(\sin ky, \color{blue}{\sin kx}\right)\right)\right)\right) \]

      11. hypot-lowering-hypot.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\mathsf{sin.f64}\left(th\right), \mathsf{hypot.f64}\left(\sin ky, \color{blue}{\sin kx}\right)\right)\right) \]

      12. sin-lowering-sin.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\mathsf{sin.f64}\left(th\right), \mathsf{hypot.f64}\left(\mathsf{sin.f64}\left(ky\right), \sin \color{blue}{kx}\right)\right)\right) \]

      13. sin-lowering-sin.f64 99.6%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\mathsf{sin.f64}\left(th\right), \mathsf{hypot.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{sin.f64}\left(kx\right)\right)\right)\right) \]

   3. Simplified 99.6%

      \[\leadsto \color{blue}{\sin ky \cdot \frac{\sin th}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in th around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\color{blue}{th}, \mathsf{hypot.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{sin.f64}\left(kx\right)\right)\right)\right) \]
   6. Step-by-step derivation

   7. Simplified 62.9%

      \[\leadsto \sin ky \cdot \frac{\color{blue}{th}}{\mathsf{hypot}\left(\sin ky, \sin kx\right)} \]

   if 0.00224999999999999983 < th

   1. Initial program 98.2%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Step-by-step derivation

      1. associate-*l/ N/A

         \[\leadsto \frac{\sin ky \cdot \sin th}{\color{blue}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}} \]

      2. associate-/l* N/A

         \[\leadsto \sin ky \cdot \color{blue}{\frac{\sin th}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\sin ky, \color{blue}{\left(\frac{\sin th}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}\right)}\right) \]

      4. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \left(\frac{\color{blue}{\sin th}}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}\right)\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\sin th, \color{blue}{\left(\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}\right)}\right)\right) \]

      6. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\mathsf{sin.f64}\left(th\right), \left(\sqrt{\color{blue}{{\sin kx}^{2} + {\sin ky}^{2}}}\right)\right)\right) \]

      7. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\mathsf{sin.f64}\left(th\right), \left(\sqrt{{\sin ky}^{2} + {\sin kx}^{2}}\right)\right)\right) \]

      8. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\mathsf{sin.f64}\left(th\right), \left(\sqrt{\sin ky \cdot \sin ky + {\sin kx}^{2}}\right)\right)\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\mathsf{sin.f64}\left(th\right), \left(\sqrt{\sin ky \cdot \sin ky + \sin kx \cdot \sin kx}\right)\right)\right) \]

      10. hypot-define N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\mathsf{sin.f64}\left(th\right), \left(\mathsf{hypot}\left(\sin ky, \color{blue}{\sin kx}\right)\right)\right)\right) \]

      11. hypot-lowering-hypot.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\mathsf{sin.f64}\left(th\right), \mathsf{hypot.f64}\left(\sin ky, \color{blue}{\sin kx}\right)\right)\right) \]

      12. sin-lowering-sin.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\mathsf{sin.f64}\left(th\right), \mathsf{hypot.f64}\left(\mathsf{sin.f64}\left(ky\right), \sin \color{blue}{kx}\right)\right)\right) \]

      13. sin-lowering-sin.f64 99.6%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\mathsf{sin.f64}\left(th\right), \mathsf{hypot.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{sin.f64}\left(kx\right)\right)\right)\right) \]

   3. Simplified 99.6%

      \[\leadsto \color{blue}{\sin ky \cdot \frac{\sin th}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in ky around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\mathsf{sin.f64}\left(th\right), \mathsf{hypot.f64}\left(\color{blue}{ky}, \mathsf{sin.f64}\left(kx\right)\right)\right)\right) \]
   6. Step-by-step derivation

   7. Simplified 48.1%

      \[\leadsto \sin ky \cdot \frac{\sin th}{\mathsf{hypot}\left(\color{blue}{ky}, \sin kx\right)} \]

   8. Taylor expanded in ky around 0

      \[\leadsto \mathsf{*.f64}\left(\color{blue}{ky}, \mathsf{/.f64}\left(\mathsf{sin.f64}\left(th\right), \mathsf{hypot.f64}\left(ky, \mathsf{sin.f64}\left(kx\right)\right)\right)\right) \]
   9. Step-by-step derivation

   10. Simplified 61.7%

       \[\leadsto \color{blue}{ky} \cdot \frac{\sin th}{\mathsf{hypot}\left(ky, \sin kx\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 11: 41.0% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sin ky \leq 10^{-96}:\\ \;\;\;\;\sin ky \cdot \frac{\sin th}{\sin kx}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= (sin ky) 1e-96) (* (sin ky) (/ (sin th) (sin kx))) (sin th)))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (sin(ky) <= 1e-96) {
		tmp = sin(ky) * (sin(th) / sin(kx));
	} else {
		tmp = sin(th);
	}
	return tmp;
}

Fortran

real(8) function code(kx, ky, th)
    real(8), intent (in) :: kx
    real(8), intent (in) :: ky
    real(8), intent (in) :: th
    real(8) :: tmp
    if (sin(ky) <= 1d-96) then
        tmp = sin(ky) * (sin(th) / sin(kx))
    else
        tmp = sin(th)
    end if
    code = tmp
end function

Java

public static double code(double kx, double ky, double th) {
	double tmp;
	if (Math.sin(ky) <= 1e-96) {
		tmp = Math.sin(ky) * (Math.sin(th) / Math.sin(kx));
	} else {
		tmp = Math.sin(th);
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if math.sin(ky) <= 1e-96:
		tmp = math.sin(ky) * (math.sin(th) / math.sin(kx))
	else:
		tmp = math.sin(th)
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (sin(ky) <= 1e-96)
		tmp = Float64(sin(ky) * Float64(sin(th) / sin(kx)));
	else
		tmp = sin(th);
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if (sin(ky) <= 1e-96)
		tmp = sin(ky) * (sin(th) / sin(kx));
	else
		tmp = sin(th);
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[N[Sin[ky], $MachinePrecision], 1e-96], N[(N[Sin[ky], $MachinePrecision] * N[(N[Sin[th], $MachinePrecision] / N[Sin[kx], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Sin[th], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (sin.f64 ky) < 9.9999999999999991e-97

   1. Initial program 91.0%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Step-by-step derivation

      1. associate-*l/ N/A

         \[\leadsto \frac{\sin ky \cdot \sin th}{\color{blue}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}} \]

      2. associate-/l* N/A

         \[\leadsto \sin ky \cdot \color{blue}{\frac{\sin th}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\sin ky, \color{blue}{\left(\frac{\sin th}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}\right)}\right) \]

      4. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \left(\frac{\color{blue}{\sin th}}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}\right)\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\sin th, \color{blue}{\left(\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}\right)}\right)\right) \]

      6. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\mathsf{sin.f64}\left(th\right), \left(\sqrt{\color{blue}{{\sin kx}^{2} + {\sin ky}^{2}}}\right)\right)\right) \]

      7. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\mathsf{sin.f64}\left(th\right), \left(\sqrt{{\sin ky}^{2} + {\sin kx}^{2}}\right)\right)\right) \]

      8. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\mathsf{sin.f64}\left(th\right), \left(\sqrt{\sin ky \cdot \sin ky + {\sin kx}^{2}}\right)\right)\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\mathsf{sin.f64}\left(th\right), \left(\sqrt{\sin ky \cdot \sin ky + \sin kx \cdot \sin kx}\right)\right)\right) \]

      10. hypot-define N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\mathsf{sin.f64}\left(th\right), \left(\mathsf{hypot}\left(\sin ky, \color{blue}{\sin kx}\right)\right)\right)\right) \]

      11. hypot-lowering-hypot.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\mathsf{sin.f64}\left(th\right), \mathsf{hypot.f64}\left(\sin ky, \color{blue}{\sin kx}\right)\right)\right) \]

      12. sin-lowering-sin.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\mathsf{sin.f64}\left(th\right), \mathsf{hypot.f64}\left(\mathsf{sin.f64}\left(ky\right), \sin \color{blue}{kx}\right)\right)\right) \]

      13. sin-lowering-sin.f64 99.6%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\mathsf{sin.f64}\left(th\right), \mathsf{hypot.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{sin.f64}\left(kx\right)\right)\right)\right) \]

   3. Simplified 99.6%

      \[\leadsto \color{blue}{\sin ky \cdot \frac{\sin th}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in ky around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \color{blue}{\left(\frac{\sin th}{\sin kx}\right)}\right) \]
   6. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\sin th, \color{blue}{\sin kx}\right)\right) \]

      2. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\mathsf{sin.f64}\left(th\right), \sin \color{blue}{kx}\right)\right) \]

      3. sin-lowering-sin.f64 30.0%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\mathsf{sin.f64}\left(th\right), \mathsf{sin.f64}\left(kx\right)\right)\right) \]

   7. Simplified 30.0%

      \[\leadsto \sin ky \cdot \color{blue}{\frac{\sin th}{\sin kx}} \]

   if 9.9999999999999991e-97 < (sin.f64 ky)

   1. Initial program 99.6%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Step-by-step derivation

      1. associate-*l/ N/A

         \[\leadsto \frac{\sin ky \cdot \sin th}{\color{blue}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}} \]

      2. associate-/l* N/A

         \[\leadsto \sin ky \cdot \color{blue}{\frac{\sin th}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\sin ky, \color{blue}{\left(\frac{\sin th}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}\right)}\right) \]

      4. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \left(\frac{\color{blue}{\sin th}}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}\right)\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\sin th, \color{blue}{\left(\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}\right)}\right)\right) \]

      6. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\mathsf{sin.f64}\left(th\right), \left(\sqrt{\color{blue}{{\sin kx}^{2} + {\sin ky}^{2}}}\right)\right)\right) \]

      7. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\mathsf{sin.f64}\left(th\right), \left(\sqrt{{\sin ky}^{2} + {\sin kx}^{2}}\right)\right)\right) \]

      8. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\mathsf{sin.f64}\left(th\right), \left(\sqrt{\sin ky \cdot \sin ky + {\sin kx}^{2}}\right)\right)\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\mathsf{sin.f64}\left(th\right), \left(\sqrt{\sin ky \cdot \sin ky + \sin kx \cdot \sin kx}\right)\right)\right) \]

      10. hypot-define N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\mathsf{sin.f64}\left(th\right), \left(\mathsf{hypot}\left(\sin ky, \color{blue}{\sin kx}\right)\right)\right)\right) \]

      11. hypot-lowering-hypot.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\mathsf{sin.f64}\left(th\right), \mathsf{hypot.f64}\left(\sin ky, \color{blue}{\sin kx}\right)\right)\right) \]

      12. sin-lowering-sin.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\mathsf{sin.f64}\left(th\right), \mathsf{hypot.f64}\left(\mathsf{sin.f64}\left(ky\right), \sin \color{blue}{kx}\right)\right)\right) \]

      13. sin-lowering-sin.f64 99.5%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\mathsf{sin.f64}\left(th\right), \mathsf{hypot.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{sin.f64}\left(kx\right)\right)\right)\right) \]

   3. Simplified 99.5%

      \[\leadsto \color{blue}{\sin ky \cdot \frac{\sin th}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in kx around 0

      \[\leadsto \color{blue}{\sin th} \]
   6. Step-by-step derivation

      1. sin-lowering-sin.f64 58.1%

         \[\leadsto \mathsf{sin.f64}\left(th\right) \]

   7. Simplified 58.1%

      \[\leadsto \color{blue}{\sin th} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 12: 33.2% accurate, 2.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sin ky \leq 10^{-96}:\\ \;\;\;\;\sin ky \cdot \frac{\sin th}{kx}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= (sin ky) 1e-96) (* (sin ky) (/ (sin th) kx)) (sin th)))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (sin(ky) <= 1e-96) {
		tmp = sin(ky) * (sin(th) / kx);
	} else {
		tmp = sin(th);
	}
	return tmp;
}

Fortran

real(8) function code(kx, ky, th)
    real(8), intent (in) :: kx
    real(8), intent (in) :: ky
    real(8), intent (in) :: th
    real(8) :: tmp
    if (sin(ky) <= 1d-96) then
        tmp = sin(ky) * (sin(th) / kx)
    else
        tmp = sin(th)
    end if
    code = tmp
end function

Java

public static double code(double kx, double ky, double th) {
	double tmp;
	if (Math.sin(ky) <= 1e-96) {
		tmp = Math.sin(ky) * (Math.sin(th) / kx);
	} else {
		tmp = Math.sin(th);
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if math.sin(ky) <= 1e-96:
		tmp = math.sin(ky) * (math.sin(th) / kx)
	else:
		tmp = math.sin(th)
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (sin(ky) <= 1e-96)
		tmp = Float64(sin(ky) * Float64(sin(th) / kx));
	else
		tmp = sin(th);
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if (sin(ky) <= 1e-96)
		tmp = sin(ky) * (sin(th) / kx);
	else
		tmp = sin(th);
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[N[Sin[ky], $MachinePrecision], 1e-96], N[(N[Sin[ky], $MachinePrecision] * N[(N[Sin[th], $MachinePrecision] / kx), $MachinePrecision]), $MachinePrecision], N[Sin[th], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (sin.f64 ky) < 9.9999999999999991e-97

   1. Initial program 91.0%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing

   3. Taylor expanded in ky around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sin.f64}\left(ky\right), \color{blue}{\sin kx}\right), \mathsf{sin.f64}\left(th\right)\right) \]
   4. Step-by-step derivation

      1. sin-lowering-sin.f64 30.0%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{sin.f64}\left(kx\right)\right), \mathsf{sin.f64}\left(th\right)\right) \]

   5. Simplified 30.0%

      \[\leadsto \frac{\sin ky}{\color{blue}{\sin kx}} \cdot \sin th \]

   6. Taylor expanded in kx around 0

      \[\leadsto \color{blue}{\frac{\sin ky \cdot \sin th}{kx}} \]
   7. Step-by-step derivation

      1. associate-/l* N/A

         \[\leadsto \sin ky \cdot \color{blue}{\frac{\sin th}{kx}} \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\sin ky, \color{blue}{\left(\frac{\sin th}{kx}\right)}\right) \]

      3. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \left(\frac{\color{blue}{\sin th}}{kx}\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\sin th, \color{blue}{kx}\right)\right) \]

      5. sin-lowering-sin.f64 19.8%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\mathsf{sin.f64}\left(th\right), kx\right)\right) \]

   8. Simplified 19.8%

      \[\leadsto \color{blue}{\sin ky \cdot \frac{\sin th}{kx}} \]

   if 9.9999999999999991e-97 < (sin.f64 ky)

   1. Initial program 99.6%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Step-by-step derivation

      1. associate-*l/ N/A

         \[\leadsto \frac{\sin ky \cdot \sin th}{\color{blue}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}} \]

      2. associate-/l* N/A

         \[\leadsto \sin ky \cdot \color{blue}{\frac{\sin th}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\sin ky, \color{blue}{\left(\frac{\sin th}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}\right)}\right) \]

      4. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \left(\frac{\color{blue}{\sin th}}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}\right)\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\sin th, \color{blue}{\left(\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}\right)}\right)\right) \]

      6. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\mathsf{sin.f64}\left(th\right), \left(\sqrt{\color{blue}{{\sin kx}^{2} + {\sin ky}^{2}}}\right)\right)\right) \]

      7. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\mathsf{sin.f64}\left(th\right), \left(\sqrt{{\sin ky}^{2} + {\sin kx}^{2}}\right)\right)\right) \]

      8. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\mathsf{sin.f64}\left(th\right), \left(\sqrt{\sin ky \cdot \sin ky + {\sin kx}^{2}}\right)\right)\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\mathsf{sin.f64}\left(th\right), \left(\sqrt{\sin ky \cdot \sin ky + \sin kx \cdot \sin kx}\right)\right)\right) \]

      10. hypot-define N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\mathsf{sin.f64}\left(th\right), \left(\mathsf{hypot}\left(\sin ky, \color{blue}{\sin kx}\right)\right)\right)\right) \]

      11. hypot-lowering-hypot.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\mathsf{sin.f64}\left(th\right), \mathsf{hypot.f64}\left(\sin ky, \color{blue}{\sin kx}\right)\right)\right) \]

      12. sin-lowering-sin.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\mathsf{sin.f64}\left(th\right), \mathsf{hypot.f64}\left(\mathsf{sin.f64}\left(ky\right), \sin \color{blue}{kx}\right)\right)\right) \]

      13. sin-lowering-sin.f64 99.5%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\mathsf{sin.f64}\left(th\right), \mathsf{hypot.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{sin.f64}\left(kx\right)\right)\right)\right) \]

   3. Simplified 99.5%

      \[\leadsto \color{blue}{\sin ky \cdot \frac{\sin th}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in kx around 0

      \[\leadsto \color{blue}{\sin th} \]
   6. Step-by-step derivation

      1. sin-lowering-sin.f64 58.1%

         \[\leadsto \mathsf{sin.f64}\left(th\right) \]

   7. Simplified 58.1%

      \[\leadsto \color{blue}{\sin th} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 13: 33.0% accurate, 3.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sin ky \leq 10^{-96}:\\ \;\;\;\;\frac{\sin th}{kx \cdot \left(\frac{-0.16666666666666666 \cdot \left(kx \cdot kx\right)}{ky} + \frac{1}{ky}\right)}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= (sin ky) 1e-96)
   (/ (sin th) (* kx (+ (/ (* -0.16666666666666666 (* kx kx)) ky) (/ 1.0 ky))))
   (sin th)))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (sin(ky) <= 1e-96) {
		tmp = sin(th) / (kx * (((-0.16666666666666666 * (kx * kx)) / ky) + (1.0 / ky)));
	} else {
		tmp = sin(th);
	}
	return tmp;
}

Fortran

real(8) function code(kx, ky, th)
    real(8), intent (in) :: kx
    real(8), intent (in) :: ky
    real(8), intent (in) :: th
    real(8) :: tmp
    if (sin(ky) <= 1d-96) then
        tmp = sin(th) / (kx * ((((-0.16666666666666666d0) * (kx * kx)) / ky) + (1.0d0 / ky)))
    else
        tmp = sin(th)
    end if
    code = tmp
end function

Java

public static double code(double kx, double ky, double th) {
	double tmp;
	if (Math.sin(ky) <= 1e-96) {
		tmp = Math.sin(th) / (kx * (((-0.16666666666666666 * (kx * kx)) / ky) + (1.0 / ky)));
	} else {
		tmp = Math.sin(th);
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if math.sin(ky) <= 1e-96:
		tmp = math.sin(th) / (kx * (((-0.16666666666666666 * (kx * kx)) / ky) + (1.0 / ky)))
	else:
		tmp = math.sin(th)
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (sin(ky) <= 1e-96)
		tmp = Float64(sin(th) / Float64(kx * Float64(Float64(Float64(-0.16666666666666666 * Float64(kx * kx)) / ky) + Float64(1.0 / ky))));
	else
		tmp = sin(th);
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if (sin(ky) <= 1e-96)
		tmp = sin(th) / (kx * (((-0.16666666666666666 * (kx * kx)) / ky) + (1.0 / ky)));
	else
		tmp = sin(th);
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[N[Sin[ky], $MachinePrecision], 1e-96], N[(N[Sin[th], $MachinePrecision] / N[(kx * N[(N[(N[(-0.16666666666666666 * N[(kx * kx), $MachinePrecision]), $MachinePrecision] / ky), $MachinePrecision] + N[(1.0 / ky), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Sin[th], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (sin.f64 ky) < 9.9999999999999991e-97

   1. Initial program 91.0%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing

   3. Taylor expanded in kx around inf

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sin.f64}\left(ky\right), \color{blue}{\left(\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}\right)}\right), \mathsf{sin.f64}\left(th\right)\right) \]
   4. Step-by-step derivation

      1. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sin.f64}\left(ky\right), \left(\sqrt{\sin kx \cdot \sin kx + {\sin ky}^{2}}\right)\right), \mathsf{sin.f64}\left(th\right)\right) \]

      2. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sin.f64}\left(ky\right), \left(\sqrt{\sin kx \cdot \sin kx + \sin ky \cdot \sin ky}\right)\right), \mathsf{sin.f64}\left(th\right)\right) \]

      3. hypot-define N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sin.f64}\left(ky\right), \left(\mathsf{hypot}\left(\sin kx, \sin ky\right)\right)\right), \mathsf{sin.f64}\left(th\right)\right) \]

      4. hypot-lowering-hypot.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{hypot.f64}\left(\sin kx, \sin ky\right)\right), \mathsf{sin.f64}\left(th\right)\right) \]

      5. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{hypot.f64}\left(\mathsf{sin.f64}\left(kx\right), \sin ky\right)\right), \mathsf{sin.f64}\left(th\right)\right) \]

      6. sin-lowering-sin.f64 99.7%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{hypot.f64}\left(\mathsf{sin.f64}\left(kx\right), \mathsf{sin.f64}\left(ky\right)\right)\right), \mathsf{sin.f64}\left(th\right)\right) \]

   5. Simplified 99.7%

      \[\leadsto \frac{\sin ky}{\color{blue}{\mathsf{hypot}\left(\sin kx, \sin ky\right)}} \cdot \sin th \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \sin th \cdot \color{blue}{\frac{\sin ky}{\sqrt{\sin kx \cdot \sin kx + \sin ky \cdot \sin ky}}} \]

      2. clear-num N/A

         \[\leadsto \sin th \cdot \frac{1}{\color{blue}{\frac{\sqrt{\sin kx \cdot \sin kx + \sin ky \cdot \sin ky}}{\sin ky}}} \]

      3. un-div-inv N/A

         \[\leadsto \frac{\sin th}{\color{blue}{\frac{\sqrt{\sin kx \cdot \sin kx + \sin ky \cdot \sin ky}}{\sin ky}}} \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\sin th, \color{blue}{\left(\frac{\sqrt{\sin kx \cdot \sin kx + \sin ky \cdot \sin ky}}{\sin ky}\right)}\right) \]

      5. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sin.f64}\left(th\right), \left(\frac{\color{blue}{\sqrt{\sin kx \cdot \sin kx + \sin ky \cdot \sin ky}}}{\sin ky}\right)\right) \]

      6. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sin.f64}\left(th\right), \left(\frac{\sqrt{\sin ky \cdot \sin ky + \sin kx \cdot \sin kx}}{\sin ky}\right)\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sin.f64}\left(th\right), \mathsf{/.f64}\left(\left(\sqrt{\sin ky \cdot \sin ky + \sin kx \cdot \sin kx}\right), \color{blue}{\sin ky}\right)\right) \]

      8. hypot-define N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sin.f64}\left(th\right), \mathsf{/.f64}\left(\left(\mathsf{hypot}\left(\sin ky, \sin kx\right)\right), \sin \color{blue}{ky}\right)\right) \]

      9. hypot-lowering-hypot.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sin.f64}\left(th\right), \mathsf{/.f64}\left(\mathsf{hypot.f64}\left(\sin ky, \sin kx\right), \sin \color{blue}{ky}\right)\right) \]

      10. sin-lowering-sin.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{sin.f64}\left(th\right), \mathsf{/.f64}\left(\mathsf{hypot.f64}\left(\mathsf{sin.f64}\left(ky\right), \sin kx\right), \sin ky\right)\right) \]

      11. sin-lowering-sin.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{sin.f64}\left(th\right), \mathsf{/.f64}\left(\mathsf{hypot.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{sin.f64}\left(kx\right)\right), \sin ky\right)\right) \]

      12. sin-lowering-sin.f64 99.6%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{sin.f64}\left(th\right), \mathsf{/.f64}\left(\mathsf{hypot.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{sin.f64}\left(kx\right)\right), \mathsf{sin.f64}\left(ky\right)\right)\right) \]

   7. Applied egg-rr 99.6%

      \[\leadsto \color{blue}{\frac{\sin th}{\frac{\mathsf{hypot}\left(\sin ky, \sin kx\right)}{\sin ky}}} \]

   8. Taylor expanded in ky around 0

      \[\leadsto \mathsf{/.f64}\left(\mathsf{sin.f64}\left(th\right), \color{blue}{\left(\frac{\sin kx}{ky}\right)}\right) \]
   9. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sin.f64}\left(th\right), \mathsf{/.f64}\left(\sin kx, \color{blue}{ky}\right)\right) \]

      2. sin-lowering-sin.f64 28.5%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sin.f64}\left(th\right), \mathsf{/.f64}\left(\mathsf{sin.f64}\left(kx\right), ky\right)\right) \]

   10. Simplified 28.5%

       \[\leadsto \frac{\sin th}{\color{blue}{\frac{\sin kx}{ky}}} \]

   11. Taylor expanded in kx around 0

       \[\leadsto \mathsf{/.f64}\left(\mathsf{sin.f64}\left(th\right), \color{blue}{\left(kx \cdot \left(\frac{-1}{6} \cdot \frac{{kx}^{2}}{ky} + \frac{1}{ky}\right)\right)}\right) \]
   12. Step-by-step derivation

       1. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{sin.f64}\left(th\right), \mathsf{*.f64}\left(kx, \color{blue}{\left(\frac{-1}{6} \cdot \frac{{kx}^{2}}{ky} + \frac{1}{ky}\right)}\right)\right) \]

       2. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{sin.f64}\left(th\right), \mathsf{*.f64}\left(kx, \mathsf{+.f64}\left(\left(\frac{-1}{6} \cdot \frac{{kx}^{2}}{ky}\right), \color{blue}{\left(\frac{1}{ky}\right)}\right)\right)\right) \]

       3. associate-*r/ N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{sin.f64}\left(th\right), \mathsf{*.f64}\left(kx, \mathsf{+.f64}\left(\left(\frac{\frac{-1}{6} \cdot {kx}^{2}}{ky}\right), \left(\frac{\color{blue}{1}}{ky}\right)\right)\right)\right) \]

       4. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{sin.f64}\left(th\right), \mathsf{*.f64}\left(kx, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\left(\frac{-1}{6} \cdot {kx}^{2}\right), ky\right), \left(\frac{\color{blue}{1}}{ky}\right)\right)\right)\right) \]

       5. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{sin.f64}\left(th\right), \mathsf{*.f64}\left(kx, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{-1}{6}, \left({kx}^{2}\right)\right), ky\right), \left(\frac{1}{ky}\right)\right)\right)\right) \]

       6. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{sin.f64}\left(th\right), \mathsf{*.f64}\left(kx, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{-1}{6}, \left(kx \cdot kx\right)\right), ky\right), \left(\frac{1}{ky}\right)\right)\right)\right) \]

       7. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{sin.f64}\left(th\right), \mathsf{*.f64}\left(kx, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(kx, kx\right)\right), ky\right), \left(\frac{1}{ky}\right)\right)\right)\right) \]

       8. /-lowering-/.f64 19.5%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{sin.f64}\left(th\right), \mathsf{*.f64}\left(kx, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(kx, kx\right)\right), ky\right), \mathsf{/.f64}\left(1, \color{blue}{ky}\right)\right)\right)\right) \]

   13. Simplified 19.5%

       \[\leadsto \frac{\sin th}{\color{blue}{kx \cdot \left(\frac{-0.16666666666666666 \cdot \left(kx \cdot kx\right)}{ky} + \frac{1}{ky}\right)}} \]

   if 9.9999999999999991e-97 < (sin.f64 ky)

   1. Initial program 99.6%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Step-by-step derivation

      1. associate-*l/ N/A

         \[\leadsto \frac{\sin ky \cdot \sin th}{\color{blue}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}} \]

      2. associate-/l* N/A

         \[\leadsto \sin ky \cdot \color{blue}{\frac{\sin th}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\sin ky, \color{blue}{\left(\frac{\sin th}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}\right)}\right) \]

      4. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \left(\frac{\color{blue}{\sin th}}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}\right)\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\sin th, \color{blue}{\left(\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}\right)}\right)\right) \]

      6. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\mathsf{sin.f64}\left(th\right), \left(\sqrt{\color{blue}{{\sin kx}^{2} + {\sin ky}^{2}}}\right)\right)\right) \]

      7. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\mathsf{sin.f64}\left(th\right), \left(\sqrt{{\sin ky}^{2} + {\sin kx}^{2}}\right)\right)\right) \]

      8. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\mathsf{sin.f64}\left(th\right), \left(\sqrt{\sin ky \cdot \sin ky + {\sin kx}^{2}}\right)\right)\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\mathsf{sin.f64}\left(th\right), \left(\sqrt{\sin ky \cdot \sin ky + \sin kx \cdot \sin kx}\right)\right)\right) \]

      10. hypot-define N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\mathsf{sin.f64}\left(th\right), \left(\mathsf{hypot}\left(\sin ky, \color{blue}{\sin kx}\right)\right)\right)\right) \]

      11. hypot-lowering-hypot.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\mathsf{sin.f64}\left(th\right), \mathsf{hypot.f64}\left(\sin ky, \color{blue}{\sin kx}\right)\right)\right) \]

      12. sin-lowering-sin.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\mathsf{sin.f64}\left(th\right), \mathsf{hypot.f64}\left(\mathsf{sin.f64}\left(ky\right), \sin \color{blue}{kx}\right)\right)\right) \]

      13. sin-lowering-sin.f64 99.5%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\mathsf{sin.f64}\left(th\right), \mathsf{hypot.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{sin.f64}\left(kx\right)\right)\right)\right) \]

   3. Simplified 99.5%

      \[\leadsto \color{blue}{\sin ky \cdot \frac{\sin th}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in kx around 0

      \[\leadsto \color{blue}{\sin th} \]
   6. Step-by-step derivation

      1. sin-lowering-sin.f64 58.1%

         \[\leadsto \mathsf{sin.f64}\left(th\right) \]

   7. Simplified 58.1%

      \[\leadsto \color{blue}{\sin th} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 14: 33.0% accurate, 3.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sin ky \leq 10^{-96}:\\ \;\;\;\;\sin th \cdot \frac{\frac{ky}{kx}}{1 + -0.16666666666666666 \cdot \left(kx \cdot kx\right)}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= (sin ky) 1e-96)
   (* (sin th) (/ (/ ky kx) (+ 1.0 (* -0.16666666666666666 (* kx kx)))))
   (sin th)))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (sin(ky) <= 1e-96) {
		tmp = sin(th) * ((ky / kx) / (1.0 + (-0.16666666666666666 * (kx * kx))));
	} else {
		tmp = sin(th);
	}
	return tmp;
}

Fortran

real(8) function code(kx, ky, th)
    real(8), intent (in) :: kx
    real(8), intent (in) :: ky
    real(8), intent (in) :: th
    real(8) :: tmp
    if (sin(ky) <= 1d-96) then
        tmp = sin(th) * ((ky / kx) / (1.0d0 + ((-0.16666666666666666d0) * (kx * kx))))
    else
        tmp = sin(th)
    end if
    code = tmp
end function

Java

public static double code(double kx, double ky, double th) {
	double tmp;
	if (Math.sin(ky) <= 1e-96) {
		tmp = Math.sin(th) * ((ky / kx) / (1.0 + (-0.16666666666666666 * (kx * kx))));
	} else {
		tmp = Math.sin(th);
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if math.sin(ky) <= 1e-96:
		tmp = math.sin(th) * ((ky / kx) / (1.0 + (-0.16666666666666666 * (kx * kx))))
	else:
		tmp = math.sin(th)
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (sin(ky) <= 1e-96)
		tmp = Float64(sin(th) * Float64(Float64(ky / kx) / Float64(1.0 + Float64(-0.16666666666666666 * Float64(kx * kx)))));
	else
		tmp = sin(th);
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if (sin(ky) <= 1e-96)
		tmp = sin(th) * ((ky / kx) / (1.0 + (-0.16666666666666666 * (kx * kx))));
	else
		tmp = sin(th);
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[N[Sin[ky], $MachinePrecision], 1e-96], N[(N[Sin[th], $MachinePrecision] * N[(N[(ky / kx), $MachinePrecision] / N[(1.0 + N[(-0.16666666666666666 * N[(kx * kx), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Sin[th], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (sin.f64 ky) < 9.9999999999999991e-97

   1. Initial program 91.0%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing

   3. Taylor expanded in kx around inf

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sin.f64}\left(ky\right), \color{blue}{\left(\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}\right)}\right), \mathsf{sin.f64}\left(th\right)\right) \]
   4. Step-by-step derivation

      1. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sin.f64}\left(ky\right), \left(\sqrt{\sin kx \cdot \sin kx + {\sin ky}^{2}}\right)\right), \mathsf{sin.f64}\left(th\right)\right) \]

      2. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sin.f64}\left(ky\right), \left(\sqrt{\sin kx \cdot \sin kx + \sin ky \cdot \sin ky}\right)\right), \mathsf{sin.f64}\left(th\right)\right) \]

      3. hypot-define N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sin.f64}\left(ky\right), \left(\mathsf{hypot}\left(\sin kx, \sin ky\right)\right)\right), \mathsf{sin.f64}\left(th\right)\right) \]

      4. hypot-lowering-hypot.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{hypot.f64}\left(\sin kx, \sin ky\right)\right), \mathsf{sin.f64}\left(th\right)\right) \]

      5. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{hypot.f64}\left(\mathsf{sin.f64}\left(kx\right), \sin ky\right)\right), \mathsf{sin.f64}\left(th\right)\right) \]

      6. sin-lowering-sin.f64 99.7%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{hypot.f64}\left(\mathsf{sin.f64}\left(kx\right), \mathsf{sin.f64}\left(ky\right)\right)\right), \mathsf{sin.f64}\left(th\right)\right) \]

   5. Simplified 99.7%

      \[\leadsto \frac{\sin ky}{\color{blue}{\mathsf{hypot}\left(\sin kx, \sin ky\right)}} \cdot \sin th \]

   6. Taylor expanded in kx around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{hypot.f64}\left(\color{blue}{\left(kx \cdot \left(1 + \frac{-1}{6} \cdot {kx}^{2}\right)\right)}, \mathsf{sin.f64}\left(ky\right)\right)\right), \mathsf{sin.f64}\left(th\right)\right) \]
   7. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(kx, \left(1 + \frac{-1}{6} \cdot {kx}^{2}\right)\right), \mathsf{sin.f64}\left(ky\right)\right)\right), \mathsf{sin.f64}\left(th\right)\right) \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(kx, \mathsf{+.f64}\left(1, \left(\frac{-1}{6} \cdot {kx}^{2}\right)\right)\right), \mathsf{sin.f64}\left(ky\right)\right)\right), \mathsf{sin.f64}\left(th\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(kx, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{6}, \left({kx}^{2}\right)\right)\right)\right), \mathsf{sin.f64}\left(ky\right)\right)\right), \mathsf{sin.f64}\left(th\right)\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(kx, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{6}, \left(kx \cdot kx\right)\right)\right)\right), \mathsf{sin.f64}\left(ky\right)\right)\right), \mathsf{sin.f64}\left(th\right)\right) \]

      5. *-lowering-*.f64 61.1%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(kx, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(kx, kx\right)\right)\right)\right), \mathsf{sin.f64}\left(ky\right)\right)\right), \mathsf{sin.f64}\left(th\right)\right) \]

   8. Simplified 61.1%

      \[\leadsto \frac{\sin ky}{\mathsf{hypot}\left(\color{blue}{kx \cdot \left(1 + -0.16666666666666666 \cdot \left(kx \cdot kx\right)\right)}, \sin ky\right)} \cdot \sin th \]

   9. Taylor expanded in ky around 0

      \[\leadsto \mathsf{*.f64}\left(\color{blue}{\left(\frac{ky}{kx \cdot \left(1 + \frac{-1}{6} \cdot {kx}^{2}\right)}\right)}, \mathsf{sin.f64}\left(th\right)\right) \]
   10. Step-by-step derivation

       1. associate-/r* N/A

          \[\leadsto \mathsf{*.f64}\left(\left(\frac{\frac{ky}{kx}}{1 + \frac{-1}{6} \cdot {kx}^{2}}\right), \mathsf{sin.f64}\left(\color{blue}{th}\right)\right) \]

       2. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\frac{ky}{kx}\right), \left(1 + \frac{-1}{6} \cdot {kx}^{2}\right)\right), \mathsf{sin.f64}\left(\color{blue}{th}\right)\right) \]

       3. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(ky, kx\right), \left(1 + \frac{-1}{6} \cdot {kx}^{2}\right)\right), \mathsf{sin.f64}\left(th\right)\right) \]

       4. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(ky, kx\right), \mathsf{+.f64}\left(1, \left(\frac{-1}{6} \cdot {kx}^{2}\right)\right)\right), \mathsf{sin.f64}\left(th\right)\right) \]

       5. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(ky, kx\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{6}, \left({kx}^{2}\right)\right)\right)\right), \mathsf{sin.f64}\left(th\right)\right) \]

       6. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(ky, kx\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{6}, \left(kx \cdot kx\right)\right)\right)\right), \mathsf{sin.f64}\left(th\right)\right) \]

       7. *-lowering-*.f64 19.5%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(ky, kx\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(kx, kx\right)\right)\right)\right), \mathsf{sin.f64}\left(th\right)\right) \]

   11. Simplified 19.5%

       \[\leadsto \color{blue}{\frac{\frac{ky}{kx}}{1 + -0.16666666666666666 \cdot \left(kx \cdot kx\right)}} \cdot \sin th \]

   if 9.9999999999999991e-97 < (sin.f64 ky)

   1. Initial program 99.6%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Step-by-step derivation

      1. associate-*l/ N/A

         \[\leadsto \frac{\sin ky \cdot \sin th}{\color{blue}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}} \]

      2. associate-/l* N/A

         \[\leadsto \sin ky \cdot \color{blue}{\frac{\sin th}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\sin ky, \color{blue}{\left(\frac{\sin th}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}\right)}\right) \]

      4. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \left(\frac{\color{blue}{\sin th}}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}\right)\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\sin th, \color{blue}{\left(\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}\right)}\right)\right) \]

      6. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\mathsf{sin.f64}\left(th\right), \left(\sqrt{\color{blue}{{\sin kx}^{2} + {\sin ky}^{2}}}\right)\right)\right) \]

      7. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\mathsf{sin.f64}\left(th\right), \left(\sqrt{{\sin ky}^{2} + {\sin kx}^{2}}\right)\right)\right) \]

      8. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\mathsf{sin.f64}\left(th\right), \left(\sqrt{\sin ky \cdot \sin ky + {\sin kx}^{2}}\right)\right)\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\mathsf{sin.f64}\left(th\right), \left(\sqrt{\sin ky \cdot \sin ky + \sin kx \cdot \sin kx}\right)\right)\right) \]

      10. hypot-define N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\mathsf{sin.f64}\left(th\right), \left(\mathsf{hypot}\left(\sin ky, \color{blue}{\sin kx}\right)\right)\right)\right) \]

      11. hypot-lowering-hypot.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\mathsf{sin.f64}\left(th\right), \mathsf{hypot.f64}\left(\sin ky, \color{blue}{\sin kx}\right)\right)\right) \]

      12. sin-lowering-sin.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\mathsf{sin.f64}\left(th\right), \mathsf{hypot.f64}\left(\mathsf{sin.f64}\left(ky\right), \sin \color{blue}{kx}\right)\right)\right) \]

      13. sin-lowering-sin.f64 99.5%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\mathsf{sin.f64}\left(th\right), \mathsf{hypot.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{sin.f64}\left(kx\right)\right)\right)\right) \]

   3. Simplified 99.5%

      \[\leadsto \color{blue}{\sin ky \cdot \frac{\sin th}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in kx around 0

      \[\leadsto \color{blue}{\sin th} \]
   6. Step-by-step derivation

      1. sin-lowering-sin.f64 58.1%

         \[\leadsto \mathsf{sin.f64}\left(th\right) \]

   7. Simplified 58.1%

      \[\leadsto \color{blue}{\sin th} \]
3. Recombined 2 regimes into one program.

4. Final simplification 31.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sin ky \leq 10^{-96}:\\ \;\;\;\;\sin th \cdot \frac{\frac{ky}{kx}}{1 + -0.16666666666666666 \cdot \left(kx \cdot kx\right)}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 33.1% accurate, 3.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;ky \leq 6 \cdot 10^{-96}:\\ \;\;\;\;ky \cdot \frac{\sin th}{\sin kx}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= ky 6e-96) (* ky (/ (sin th) (sin kx))) (sin th)))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (ky <= 6e-96) {
		tmp = ky * (sin(th) / sin(kx));
	} else {
		tmp = sin(th);
	}
	return tmp;
}

Fortran

real(8) function code(kx, ky, th)
    real(8), intent (in) :: kx
    real(8), intent (in) :: ky
    real(8), intent (in) :: th
    real(8) :: tmp
    if (ky <= 6d-96) then
        tmp = ky * (sin(th) / sin(kx))
    else
        tmp = sin(th)
    end if
    code = tmp
end function

Java

public static double code(double kx, double ky, double th) {
	double tmp;
	if (ky <= 6e-96) {
		tmp = ky * (Math.sin(th) / Math.sin(kx));
	} else {
		tmp = Math.sin(th);
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if ky <= 6e-96:
		tmp = ky * (math.sin(th) / math.sin(kx))
	else:
		tmp = math.sin(th)
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (ky <= 6e-96)
		tmp = Float64(ky * Float64(sin(th) / sin(kx)));
	else
		tmp = sin(th);
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if (ky <= 6e-96)
		tmp = ky * (sin(th) / sin(kx));
	else
		tmp = sin(th);
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[ky, 6e-96], N[(ky * N[(N[Sin[th], $MachinePrecision] / N[Sin[kx], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Sin[th], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if ky < 6e-96

   1. Initial program 91.2%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing

   3. Taylor expanded in kx around inf

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sin.f64}\left(ky\right), \color{blue}{\left(\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}\right)}\right), \mathsf{sin.f64}\left(th\right)\right) \]
   4. Step-by-step derivation

      1. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sin.f64}\left(ky\right), \left(\sqrt{\sin kx \cdot \sin kx + {\sin ky}^{2}}\right)\right), \mathsf{sin.f64}\left(th\right)\right) \]

      2. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sin.f64}\left(ky\right), \left(\sqrt{\sin kx \cdot \sin kx + \sin ky \cdot \sin ky}\right)\right), \mathsf{sin.f64}\left(th\right)\right) \]

      3. hypot-define N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sin.f64}\left(ky\right), \left(\mathsf{hypot}\left(\sin kx, \sin ky\right)\right)\right), \mathsf{sin.f64}\left(th\right)\right) \]

      4. hypot-lowering-hypot.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{hypot.f64}\left(\sin kx, \sin ky\right)\right), \mathsf{sin.f64}\left(th\right)\right) \]

      5. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{hypot.f64}\left(\mathsf{sin.f64}\left(kx\right), \sin ky\right)\right), \mathsf{sin.f64}\left(th\right)\right) \]

      6. sin-lowering-sin.f64 99.7%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{hypot.f64}\left(\mathsf{sin.f64}\left(kx\right), \mathsf{sin.f64}\left(ky\right)\right)\right), \mathsf{sin.f64}\left(th\right)\right) \]

   5. Simplified 99.7%

      \[\leadsto \frac{\sin ky}{\color{blue}{\mathsf{hypot}\left(\sin kx, \sin ky\right)}} \cdot \sin th \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \sin th \cdot \color{blue}{\frac{\sin ky}{\sqrt{\sin kx \cdot \sin kx + \sin ky \cdot \sin ky}}} \]

      2. clear-num N/A

         \[\leadsto \sin th \cdot \frac{1}{\color{blue}{\frac{\sqrt{\sin kx \cdot \sin kx + \sin ky \cdot \sin ky}}{\sin ky}}} \]

      3. un-div-inv N/A

         \[\leadsto \frac{\sin th}{\color{blue}{\frac{\sqrt{\sin kx \cdot \sin kx + \sin ky \cdot \sin ky}}{\sin ky}}} \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\sin th, \color{blue}{\left(\frac{\sqrt{\sin kx \cdot \sin kx + \sin ky \cdot \sin ky}}{\sin ky}\right)}\right) \]

      5. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sin.f64}\left(th\right), \left(\frac{\color{blue}{\sqrt{\sin kx \cdot \sin kx + \sin ky \cdot \sin ky}}}{\sin ky}\right)\right) \]

      6. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sin.f64}\left(th\right), \left(\frac{\sqrt{\sin ky \cdot \sin ky + \sin kx \cdot \sin kx}}{\sin ky}\right)\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sin.f64}\left(th\right), \mathsf{/.f64}\left(\left(\sqrt{\sin ky \cdot \sin ky + \sin kx \cdot \sin kx}\right), \color{blue}{\sin ky}\right)\right) \]

      8. hypot-define N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sin.f64}\left(th\right), \mathsf{/.f64}\left(\left(\mathsf{hypot}\left(\sin ky, \sin kx\right)\right), \sin \color{blue}{ky}\right)\right) \]

      9. hypot-lowering-hypot.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sin.f64}\left(th\right), \mathsf{/.f64}\left(\mathsf{hypot.f64}\left(\sin ky, \sin kx\right), \sin \color{blue}{ky}\right)\right) \]

      10. sin-lowering-sin.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{sin.f64}\left(th\right), \mathsf{/.f64}\left(\mathsf{hypot.f64}\left(\mathsf{sin.f64}\left(ky\right), \sin kx\right), \sin ky\right)\right) \]

      11. sin-lowering-sin.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{sin.f64}\left(th\right), \mathsf{/.f64}\left(\mathsf{hypot.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{sin.f64}\left(kx\right)\right), \sin ky\right)\right) \]

      12. sin-lowering-sin.f64 99.7%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{sin.f64}\left(th\right), \mathsf{/.f64}\left(\mathsf{hypot.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{sin.f64}\left(kx\right)\right), \mathsf{sin.f64}\left(ky\right)\right)\right) \]

   7. Applied egg-rr 99.7%

      \[\leadsto \color{blue}{\frac{\sin th}{\frac{\mathsf{hypot}\left(\sin ky, \sin kx\right)}{\sin ky}}} \]

   8. Taylor expanded in ky around 0

      \[\leadsto \mathsf{/.f64}\left(\mathsf{sin.f64}\left(th\right), \color{blue}{\left(\frac{\sin kx}{ky}\right)}\right) \]
   9. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sin.f64}\left(th\right), \mathsf{/.f64}\left(\sin kx, \color{blue}{ky}\right)\right) \]

      2. sin-lowering-sin.f64 27.8%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sin.f64}\left(th\right), \mathsf{/.f64}\left(\mathsf{sin.f64}\left(kx\right), ky\right)\right) \]

   10. Simplified 27.8%

       \[\leadsto \frac{\sin th}{\color{blue}{\frac{\sin kx}{ky}}} \]
   11. Step-by-step derivation

       1. associate-/r/ N/A

          \[\leadsto \frac{\sin th}{\sin kx} \cdot \color{blue}{ky} \]

       2. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\left(\frac{\sin th}{\sin kx}\right), \color{blue}{ky}\right) \]

       3. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\sin th, \sin kx\right), ky\right) \]

       4. sin-lowering-sin.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sin.f64}\left(th\right), \sin kx\right), ky\right) \]

       5. sin-lowering-sin.f64 27.9%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sin.f64}\left(th\right), \mathsf{sin.f64}\left(kx\right)\right), ky\right) \]

   12. Applied egg-rr 27.9%

       \[\leadsto \color{blue}{\frac{\sin th}{\sin kx} \cdot ky} \]

   if 6e-96 < ky

   1. Initial program 99.7%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Step-by-step derivation

      1. associate-*l/ N/A

         \[\leadsto \frac{\sin ky \cdot \sin th}{\color{blue}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}} \]

      2. associate-/l* N/A

         \[\leadsto \sin ky \cdot \color{blue}{\frac{\sin th}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\sin ky, \color{blue}{\left(\frac{\sin th}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}\right)}\right) \]

      4. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \left(\frac{\color{blue}{\sin th}}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}\right)\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\sin th, \color{blue}{\left(\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}\right)}\right)\right) \]

      6. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\mathsf{sin.f64}\left(th\right), \left(\sqrt{\color{blue}{{\sin kx}^{2} + {\sin ky}^{2}}}\right)\right)\right) \]

      7. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\mathsf{sin.f64}\left(th\right), \left(\sqrt{{\sin ky}^{2} + {\sin kx}^{2}}\right)\right)\right) \]

      8. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\mathsf{sin.f64}\left(th\right), \left(\sqrt{\sin ky \cdot \sin ky + {\sin kx}^{2}}\right)\right)\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\mathsf{sin.f64}\left(th\right), \left(\sqrt{\sin ky \cdot \sin ky + \sin kx \cdot \sin kx}\right)\right)\right) \]

      10. hypot-define N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\mathsf{sin.f64}\left(th\right), \left(\mathsf{hypot}\left(\sin ky, \color{blue}{\sin kx}\right)\right)\right)\right) \]

      11. hypot-lowering-hypot.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\mathsf{sin.f64}\left(th\right), \mathsf{hypot.f64}\left(\sin ky, \color{blue}{\sin kx}\right)\right)\right) \]

      12. sin-lowering-sin.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\mathsf{sin.f64}\left(th\right), \mathsf{hypot.f64}\left(\mathsf{sin.f64}\left(ky\right), \sin \color{blue}{kx}\right)\right)\right) \]

      13. sin-lowering-sin.f64 99.5%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\mathsf{sin.f64}\left(th\right), \mathsf{hypot.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{sin.f64}\left(kx\right)\right)\right)\right) \]

   3. Simplified 99.5%

      \[\leadsto \color{blue}{\sin ky \cdot \frac{\sin th}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in kx around 0

      \[\leadsto \color{blue}{\sin th} \]
   6. Step-by-step derivation

      1. sin-lowering-sin.f64 37.0%

         \[\leadsto \mathsf{sin.f64}\left(th\right) \]

   7. Simplified 37.0%

      \[\leadsto \color{blue}{\sin th} \]
3. Recombined 2 regimes into one program.

4. Final simplification 30.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;ky \leq 6 \cdot 10^{-96}:\\ \;\;\;\;ky \cdot \frac{\sin th}{\sin kx}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 33.1% accurate, 3.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;ky \leq 5 \cdot 10^{-96}:\\ \;\;\;\;\sin th \cdot \frac{ky}{\sin kx}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= ky 5e-96) (* (sin th) (/ ky (sin kx))) (sin th)))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (ky <= 5e-96) {
		tmp = sin(th) * (ky / sin(kx));
	} else {
		tmp = sin(th);
	}
	return tmp;
}

Fortran

real(8) function code(kx, ky, th)
    real(8), intent (in) :: kx
    real(8), intent (in) :: ky
    real(8), intent (in) :: th
    real(8) :: tmp
    if (ky <= 5d-96) then
        tmp = sin(th) * (ky / sin(kx))
    else
        tmp = sin(th)
    end if
    code = tmp
end function

Java

public static double code(double kx, double ky, double th) {
	double tmp;
	if (ky <= 5e-96) {
		tmp = Math.sin(th) * (ky / Math.sin(kx));
	} else {
		tmp = Math.sin(th);
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if ky <= 5e-96:
		tmp = math.sin(th) * (ky / math.sin(kx))
	else:
		tmp = math.sin(th)
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (ky <= 5e-96)
		tmp = Float64(sin(th) * Float64(ky / sin(kx)));
	else
		tmp = sin(th);
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if (ky <= 5e-96)
		tmp = sin(th) * (ky / sin(kx));
	else
		tmp = sin(th);
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[ky, 5e-96], N[(N[Sin[th], $MachinePrecision] * N[(ky / N[Sin[kx], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Sin[th], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if ky < 4.99999999999999995e-96

   1. Initial program 91.2%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing

   3. Taylor expanded in ky around 0

      \[\leadsto \mathsf{*.f64}\left(\color{blue}{\left(\frac{ky}{\sin kx}\right)}, \mathsf{sin.f64}\left(th\right)\right) \]
   4. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(ky, \sin kx\right), \mathsf{sin.f64}\left(\color{blue}{th}\right)\right) \]

      2. sin-lowering-sin.f64 27.8%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(ky, \mathsf{sin.f64}\left(kx\right)\right), \mathsf{sin.f64}\left(th\right)\right) \]

   5. Simplified 27.8%

      \[\leadsto \color{blue}{\frac{ky}{\sin kx}} \cdot \sin th \]

   if 4.99999999999999995e-96 < ky

   1. Initial program 99.7%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Step-by-step derivation

      1. associate-*l/ N/A

         \[\leadsto \frac{\sin ky \cdot \sin th}{\color{blue}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}} \]

      2. associate-/l* N/A

         \[\leadsto \sin ky \cdot \color{blue}{\frac{\sin th}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\sin ky, \color{blue}{\left(\frac{\sin th}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}\right)}\right) \]

      4. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \left(\frac{\color{blue}{\sin th}}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}\right)\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\sin th, \color{blue}{\left(\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}\right)}\right)\right) \]

      6. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\mathsf{sin.f64}\left(th\right), \left(\sqrt{\color{blue}{{\sin kx}^{2} + {\sin ky}^{2}}}\right)\right)\right) \]

      7. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\mathsf{sin.f64}\left(th\right), \left(\sqrt{{\sin ky}^{2} + {\sin kx}^{2}}\right)\right)\right) \]

      8. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\mathsf{sin.f64}\left(th\right), \left(\sqrt{\sin ky \cdot \sin ky + {\sin kx}^{2}}\right)\right)\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\mathsf{sin.f64}\left(th\right), \left(\sqrt{\sin ky \cdot \sin ky + \sin kx \cdot \sin kx}\right)\right)\right) \]

      10. hypot-define N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\mathsf{sin.f64}\left(th\right), \left(\mathsf{hypot}\left(\sin ky, \color{blue}{\sin kx}\right)\right)\right)\right) \]

      11. hypot-lowering-hypot.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\mathsf{sin.f64}\left(th\right), \mathsf{hypot.f64}\left(\sin ky, \color{blue}{\sin kx}\right)\right)\right) \]

      12. sin-lowering-sin.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\mathsf{sin.f64}\left(th\right), \mathsf{hypot.f64}\left(\mathsf{sin.f64}\left(ky\right), \sin \color{blue}{kx}\right)\right)\right) \]

      13. sin-lowering-sin.f64 99.5%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\mathsf{sin.f64}\left(th\right), \mathsf{hypot.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{sin.f64}\left(kx\right)\right)\right)\right) \]

   3. Simplified 99.5%

      \[\leadsto \color{blue}{\sin ky \cdot \frac{\sin th}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in kx around 0

      \[\leadsto \color{blue}{\sin th} \]
   6. Step-by-step derivation

      1. sin-lowering-sin.f64 37.0%

         \[\leadsto \mathsf{sin.f64}\left(th\right) \]

   7. Simplified 37.0%

      \[\leadsto \color{blue}{\sin th} \]
3. Recombined 2 regimes into one program.

4. Final simplification 30.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;ky \leq 5 \cdot 10^{-96}:\\ \;\;\;\;\sin th \cdot \frac{ky}{\sin kx}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 26.3% accurate, 6.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;ky \leq 8.6 \cdot 10^{-97}:\\ \;\;\;\;\frac{\sin th}{\frac{kx}{ky}}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= ky 8.6e-97) (/ (sin th) (/ kx ky)) (sin th)))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (ky <= 8.6e-97) {
		tmp = sin(th) / (kx / ky);
	} else {
		tmp = sin(th);
	}
	return tmp;
}

Fortran

real(8) function code(kx, ky, th)
    real(8), intent (in) :: kx
    real(8), intent (in) :: ky
    real(8), intent (in) :: th
    real(8) :: tmp
    if (ky <= 8.6d-97) then
        tmp = sin(th) / (kx / ky)
    else
        tmp = sin(th)
    end if
    code = tmp
end function

Java

public static double code(double kx, double ky, double th) {
	double tmp;
	if (ky <= 8.6e-97) {
		tmp = Math.sin(th) / (kx / ky);
	} else {
		tmp = Math.sin(th);
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if ky <= 8.6e-97:
		tmp = math.sin(th) / (kx / ky)
	else:
		tmp = math.sin(th)
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (ky <= 8.6e-97)
		tmp = Float64(sin(th) / Float64(kx / ky));
	else
		tmp = sin(th);
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if (ky <= 8.6e-97)
		tmp = sin(th) / (kx / ky);
	else
		tmp = sin(th);
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[ky, 8.6e-97], N[(N[Sin[th], $MachinePrecision] / N[(kx / ky), $MachinePrecision]), $MachinePrecision], N[Sin[th], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if ky < 8.6e-97

   1. Initial program 91.2%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing

   3. Taylor expanded in kx around inf

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sin.f64}\left(ky\right), \color{blue}{\left(\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}\right)}\right), \mathsf{sin.f64}\left(th\right)\right) \]
   4. Step-by-step derivation

      1. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sin.f64}\left(ky\right), \left(\sqrt{\sin kx \cdot \sin kx + {\sin ky}^{2}}\right)\right), \mathsf{sin.f64}\left(th\right)\right) \]

      2. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sin.f64}\left(ky\right), \left(\sqrt{\sin kx \cdot \sin kx + \sin ky \cdot \sin ky}\right)\right), \mathsf{sin.f64}\left(th\right)\right) \]

      3. hypot-define N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sin.f64}\left(ky\right), \left(\mathsf{hypot}\left(\sin kx, \sin ky\right)\right)\right), \mathsf{sin.f64}\left(th\right)\right) \]

      4. hypot-lowering-hypot.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{hypot.f64}\left(\sin kx, \sin ky\right)\right), \mathsf{sin.f64}\left(th\right)\right) \]

      5. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{hypot.f64}\left(\mathsf{sin.f64}\left(kx\right), \sin ky\right)\right), \mathsf{sin.f64}\left(th\right)\right) \]

      6. sin-lowering-sin.f64 99.7%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{hypot.f64}\left(\mathsf{sin.f64}\left(kx\right), \mathsf{sin.f64}\left(ky\right)\right)\right), \mathsf{sin.f64}\left(th\right)\right) \]

   5. Simplified 99.7%

      \[\leadsto \frac{\sin ky}{\color{blue}{\mathsf{hypot}\left(\sin kx, \sin ky\right)}} \cdot \sin th \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \sin th \cdot \color{blue}{\frac{\sin ky}{\sqrt{\sin kx \cdot \sin kx + \sin ky \cdot \sin ky}}} \]

      2. clear-num N/A

         \[\leadsto \sin th \cdot \frac{1}{\color{blue}{\frac{\sqrt{\sin kx \cdot \sin kx + \sin ky \cdot \sin ky}}{\sin ky}}} \]

      3. un-div-inv N/A

         \[\leadsto \frac{\sin th}{\color{blue}{\frac{\sqrt{\sin kx \cdot \sin kx + \sin ky \cdot \sin ky}}{\sin ky}}} \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\sin th, \color{blue}{\left(\frac{\sqrt{\sin kx \cdot \sin kx + \sin ky \cdot \sin ky}}{\sin ky}\right)}\right) \]

      5. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sin.f64}\left(th\right), \left(\frac{\color{blue}{\sqrt{\sin kx \cdot \sin kx + \sin ky \cdot \sin ky}}}{\sin ky}\right)\right) \]

      6. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sin.f64}\left(th\right), \left(\frac{\sqrt{\sin ky \cdot \sin ky + \sin kx \cdot \sin kx}}{\sin ky}\right)\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sin.f64}\left(th\right), \mathsf{/.f64}\left(\left(\sqrt{\sin ky \cdot \sin ky + \sin kx \cdot \sin kx}\right), \color{blue}{\sin ky}\right)\right) \]

      8. hypot-define N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sin.f64}\left(th\right), \mathsf{/.f64}\left(\left(\mathsf{hypot}\left(\sin ky, \sin kx\right)\right), \sin \color{blue}{ky}\right)\right) \]

      9. hypot-lowering-hypot.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sin.f64}\left(th\right), \mathsf{/.f64}\left(\mathsf{hypot.f64}\left(\sin ky, \sin kx\right), \sin \color{blue}{ky}\right)\right) \]

      10. sin-lowering-sin.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{sin.f64}\left(th\right), \mathsf{/.f64}\left(\mathsf{hypot.f64}\left(\mathsf{sin.f64}\left(ky\right), \sin kx\right), \sin ky\right)\right) \]

      11. sin-lowering-sin.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{sin.f64}\left(th\right), \mathsf{/.f64}\left(\mathsf{hypot.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{sin.f64}\left(kx\right)\right), \sin ky\right)\right) \]

      12. sin-lowering-sin.f64 99.7%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{sin.f64}\left(th\right), \mathsf{/.f64}\left(\mathsf{hypot.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{sin.f64}\left(kx\right)\right), \mathsf{sin.f64}\left(ky\right)\right)\right) \]

   7. Applied egg-rr 99.7%

      \[\leadsto \color{blue}{\frac{\sin th}{\frac{\mathsf{hypot}\left(\sin ky, \sin kx\right)}{\sin ky}}} \]

   8. Taylor expanded in ky around 0

      \[\leadsto \mathsf{/.f64}\left(\mathsf{sin.f64}\left(th\right), \color{blue}{\left(\frac{\sin kx}{ky}\right)}\right) \]
   9. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sin.f64}\left(th\right), \mathsf{/.f64}\left(\sin kx, \color{blue}{ky}\right)\right) \]

      2. sin-lowering-sin.f64 27.8%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sin.f64}\left(th\right), \mathsf{/.f64}\left(\mathsf{sin.f64}\left(kx\right), ky\right)\right) \]

   10. Simplified 27.8%

       \[\leadsto \frac{\sin th}{\color{blue}{\frac{\sin kx}{ky}}} \]

   11. Taylor expanded in kx around 0

       \[\leadsto \mathsf{/.f64}\left(\mathsf{sin.f64}\left(th\right), \color{blue}{\left(\frac{kx}{ky}\right)}\right) \]
   12. Step-by-step derivation

       1. /-lowering-/.f64 19.2%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{sin.f64}\left(th\right), \mathsf{/.f64}\left(kx, \color{blue}{ky}\right)\right) \]

   13. Simplified 19.2%

       \[\leadsto \frac{\sin th}{\color{blue}{\frac{kx}{ky}}} \]

   if 8.6e-97 < ky

   1. Initial program 99.7%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Step-by-step derivation

      1. associate-*l/ N/A

         \[\leadsto \frac{\sin ky \cdot \sin th}{\color{blue}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}} \]

      2. associate-/l* N/A

         \[\leadsto \sin ky \cdot \color{blue}{\frac{\sin th}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\sin ky, \color{blue}{\left(\frac{\sin th}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}\right)}\right) \]

      4. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \left(\frac{\color{blue}{\sin th}}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}\right)\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\sin th, \color{blue}{\left(\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}\right)}\right)\right) \]

      6. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\mathsf{sin.f64}\left(th\right), \left(\sqrt{\color{blue}{{\sin kx}^{2} + {\sin ky}^{2}}}\right)\right)\right) \]

      7. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\mathsf{sin.f64}\left(th\right), \left(\sqrt{{\sin ky}^{2} + {\sin kx}^{2}}\right)\right)\right) \]

      8. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\mathsf{sin.f64}\left(th\right), \left(\sqrt{\sin ky \cdot \sin ky + {\sin kx}^{2}}\right)\right)\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\mathsf{sin.f64}\left(th\right), \left(\sqrt{\sin ky \cdot \sin ky + \sin kx \cdot \sin kx}\right)\right)\right) \]

      10. hypot-define N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\mathsf{sin.f64}\left(th\right), \left(\mathsf{hypot}\left(\sin ky, \color{blue}{\sin kx}\right)\right)\right)\right) \]

      11. hypot-lowering-hypot.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\mathsf{sin.f64}\left(th\right), \mathsf{hypot.f64}\left(\sin ky, \color{blue}{\sin kx}\right)\right)\right) \]

      12. sin-lowering-sin.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\mathsf{sin.f64}\left(th\right), \mathsf{hypot.f64}\left(\mathsf{sin.f64}\left(ky\right), \sin \color{blue}{kx}\right)\right)\right) \]

      13. sin-lowering-sin.f64 99.5%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\mathsf{sin.f64}\left(th\right), \mathsf{hypot.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{sin.f64}\left(kx\right)\right)\right)\right) \]

   3. Simplified 99.5%

      \[\leadsto \color{blue}{\sin ky \cdot \frac{\sin th}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in kx around 0

      \[\leadsto \color{blue}{\sin th} \]
   6. Step-by-step derivation

      1. sin-lowering-sin.f64 37.0%

         \[\leadsto \mathsf{sin.f64}\left(th\right) \]

   7. Simplified 37.0%

      \[\leadsto \color{blue}{\sin th} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 18: 26.9% accurate, 6.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;kx \leq 5.8 \cdot 10^{-15}:\\ \;\;\;\;\sin th\\ \mathbf{else}:\\ \;\;\;\;\frac{th}{\frac{\sin kx}{ky}}\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= kx 5.8e-15) (sin th) (/ th (/ (sin kx) ky))))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (kx <= 5.8e-15) {
		tmp = sin(th);
	} else {
		tmp = th / (sin(kx) / ky);
	}
	return tmp;
}

Fortran

real(8) function code(kx, ky, th)
    real(8), intent (in) :: kx
    real(8), intent (in) :: ky
    real(8), intent (in) :: th
    real(8) :: tmp
    if (kx <= 5.8d-15) then
        tmp = sin(th)
    else
        tmp = th / (sin(kx) / ky)
    end if
    code = tmp
end function

Java

public static double code(double kx, double ky, double th) {
	double tmp;
	if (kx <= 5.8e-15) {
		tmp = Math.sin(th);
	} else {
		tmp = th / (Math.sin(kx) / ky);
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if kx <= 5.8e-15:
		tmp = math.sin(th)
	else:
		tmp = th / (math.sin(kx) / ky)
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (kx <= 5.8e-15)
		tmp = sin(th);
	else
		tmp = Float64(th / Float64(sin(kx) / ky));
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if (kx <= 5.8e-15)
		tmp = sin(th);
	else
		tmp = th / (sin(kx) / ky);
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[kx, 5.8e-15], N[Sin[th], $MachinePrecision], N[(th / N[(N[Sin[kx], $MachinePrecision] / ky), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if kx < 5.80000000000000037e-15

   1. Initial program 92.0%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Step-by-step derivation

      1. associate-*l/ N/A

         \[\leadsto \frac{\sin ky \cdot \sin th}{\color{blue}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}} \]

      2. associate-/l* N/A

         \[\leadsto \sin ky \cdot \color{blue}{\frac{\sin th}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\sin ky, \color{blue}{\left(\frac{\sin th}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}\right)}\right) \]

      4. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \left(\frac{\color{blue}{\sin th}}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}\right)\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\sin th, \color{blue}{\left(\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}\right)}\right)\right) \]

      6. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\mathsf{sin.f64}\left(th\right), \left(\sqrt{\color{blue}{{\sin kx}^{2} + {\sin ky}^{2}}}\right)\right)\right) \]

      7. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\mathsf{sin.f64}\left(th\right), \left(\sqrt{{\sin ky}^{2} + {\sin kx}^{2}}\right)\right)\right) \]

      8. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\mathsf{sin.f64}\left(th\right), \left(\sqrt{\sin ky \cdot \sin ky + {\sin kx}^{2}}\right)\right)\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\mathsf{sin.f64}\left(th\right), \left(\sqrt{\sin ky \cdot \sin ky + \sin kx \cdot \sin kx}\right)\right)\right) \]

      10. hypot-define N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\mathsf{sin.f64}\left(th\right), \left(\mathsf{hypot}\left(\sin ky, \color{blue}{\sin kx}\right)\right)\right)\right) \]

      11. hypot-lowering-hypot.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\mathsf{sin.f64}\left(th\right), \mathsf{hypot.f64}\left(\sin ky, \color{blue}{\sin kx}\right)\right)\right) \]

      12. sin-lowering-sin.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\mathsf{sin.f64}\left(th\right), \mathsf{hypot.f64}\left(\mathsf{sin.f64}\left(ky\right), \sin \color{blue}{kx}\right)\right)\right) \]

      13. sin-lowering-sin.f64 99.6%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\mathsf{sin.f64}\left(th\right), \mathsf{hypot.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{sin.f64}\left(kx\right)\right)\right)\right) \]

   3. Simplified 99.6%

      \[\leadsto \color{blue}{\sin ky \cdot \frac{\sin th}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in kx around 0

      \[\leadsto \color{blue}{\sin th} \]
   6. Step-by-step derivation

      1. sin-lowering-sin.f64 29.8%

         \[\leadsto \mathsf{sin.f64}\left(th\right) \]

   7. Simplified 29.8%

      \[\leadsto \color{blue}{\sin th} \]

   if 5.80000000000000037e-15 < kx

   1. Initial program 99.4%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing

   3. Taylor expanded in kx around inf

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sin.f64}\left(ky\right), \color{blue}{\left(\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}\right)}\right), \mathsf{sin.f64}\left(th\right)\right) \]
   4. Step-by-step derivation

      1. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sin.f64}\left(ky\right), \left(\sqrt{\sin kx \cdot \sin kx + {\sin ky}^{2}}\right)\right), \mathsf{sin.f64}\left(th\right)\right) \]

      2. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sin.f64}\left(ky\right), \left(\sqrt{\sin kx \cdot \sin kx + \sin ky \cdot \sin ky}\right)\right), \mathsf{sin.f64}\left(th\right)\right) \]

      3. hypot-define N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sin.f64}\left(ky\right), \left(\mathsf{hypot}\left(\sin kx, \sin ky\right)\right)\right), \mathsf{sin.f64}\left(th\right)\right) \]

      4. hypot-lowering-hypot.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{hypot.f64}\left(\sin kx, \sin ky\right)\right), \mathsf{sin.f64}\left(th\right)\right) \]

      5. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{hypot.f64}\left(\mathsf{sin.f64}\left(kx\right), \sin ky\right)\right), \mathsf{sin.f64}\left(th\right)\right) \]

      6. sin-lowering-sin.f64 99.4%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{hypot.f64}\left(\mathsf{sin.f64}\left(kx\right), \mathsf{sin.f64}\left(ky\right)\right)\right), \mathsf{sin.f64}\left(th\right)\right) \]

   5. Simplified 99.4%

      \[\leadsto \frac{\sin ky}{\color{blue}{\mathsf{hypot}\left(\sin kx, \sin ky\right)}} \cdot \sin th \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \sin th \cdot \color{blue}{\frac{\sin ky}{\sqrt{\sin kx \cdot \sin kx + \sin ky \cdot \sin ky}}} \]

      2. clear-num N/A

         \[\leadsto \sin th \cdot \frac{1}{\color{blue}{\frac{\sqrt{\sin kx \cdot \sin kx + \sin ky \cdot \sin ky}}{\sin ky}}} \]

      3. un-div-inv N/A

         \[\leadsto \frac{\sin th}{\color{blue}{\frac{\sqrt{\sin kx \cdot \sin kx + \sin ky \cdot \sin ky}}{\sin ky}}} \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\sin th, \color{blue}{\left(\frac{\sqrt{\sin kx \cdot \sin kx + \sin ky \cdot \sin ky}}{\sin ky}\right)}\right) \]

      5. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sin.f64}\left(th\right), \left(\frac{\color{blue}{\sqrt{\sin kx \cdot \sin kx + \sin ky \cdot \sin ky}}}{\sin ky}\right)\right) \]

      6. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sin.f64}\left(th\right), \left(\frac{\sqrt{\sin ky \cdot \sin ky + \sin kx \cdot \sin kx}}{\sin ky}\right)\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sin.f64}\left(th\right), \mathsf{/.f64}\left(\left(\sqrt{\sin ky \cdot \sin ky + \sin kx \cdot \sin kx}\right), \color{blue}{\sin ky}\right)\right) \]

      8. hypot-define N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sin.f64}\left(th\right), \mathsf{/.f64}\left(\left(\mathsf{hypot}\left(\sin ky, \sin kx\right)\right), \sin \color{blue}{ky}\right)\right) \]

      9. hypot-lowering-hypot.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sin.f64}\left(th\right), \mathsf{/.f64}\left(\mathsf{hypot.f64}\left(\sin ky, \sin kx\right), \sin \color{blue}{ky}\right)\right) \]

      10. sin-lowering-sin.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{sin.f64}\left(th\right), \mathsf{/.f64}\left(\mathsf{hypot.f64}\left(\mathsf{sin.f64}\left(ky\right), \sin kx\right), \sin ky\right)\right) \]

      11. sin-lowering-sin.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{sin.f64}\left(th\right), \mathsf{/.f64}\left(\mathsf{hypot.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{sin.f64}\left(kx\right)\right), \sin ky\right)\right) \]

      12. sin-lowering-sin.f64 99.4%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{sin.f64}\left(th\right), \mathsf{/.f64}\left(\mathsf{hypot.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{sin.f64}\left(kx\right)\right), \mathsf{sin.f64}\left(ky\right)\right)\right) \]

   7. Applied egg-rr 99.4%

      \[\leadsto \color{blue}{\frac{\sin th}{\frac{\mathsf{hypot}\left(\sin ky, \sin kx\right)}{\sin ky}}} \]

   8. Taylor expanded in th around 0

      \[\leadsto \mathsf{/.f64}\left(\color{blue}{th}, \mathsf{/.f64}\left(\mathsf{hypot.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{sin.f64}\left(kx\right)\right), \mathsf{sin.f64}\left(ky\right)\right)\right) \]
   9. Step-by-step derivation

   10. Simplified 58.1%

       \[\leadsto \frac{\color{blue}{th}}{\frac{\mathsf{hypot}\left(\sin ky, \sin kx\right)}{\sin ky}} \]

   11. Taylor expanded in ky around 0

       \[\leadsto \mathsf{/.f64}\left(th, \color{blue}{\left(\frac{\sin kx}{ky}\right)}\right) \]
   12. Step-by-step derivation

       1. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(th, \mathsf{/.f64}\left(\sin kx, \color{blue}{ky}\right)\right) \]

       2. sin-lowering-sin.f64 22.7%

          \[\leadsto \mathsf{/.f64}\left(th, \mathsf{/.f64}\left(\mathsf{sin.f64}\left(kx\right), ky\right)\right) \]

   13. Simplified 22.7%

       \[\leadsto \frac{th}{\color{blue}{\frac{\sin kx}{ky}}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 19: 26.9% accurate, 6.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;kx \leq 5.4 \cdot 10^{-15}:\\ \;\;\;\;\sin th\\ \mathbf{else}:\\ \;\;\;\;ky \cdot \frac{th}{\sin kx}\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= kx 5.4e-15) (sin th) (* ky (/ th (sin kx)))))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (kx <= 5.4e-15) {
		tmp = sin(th);
	} else {
		tmp = ky * (th / sin(kx));
	}
	return tmp;
}

Fortran

real(8) function code(kx, ky, th)
    real(8), intent (in) :: kx
    real(8), intent (in) :: ky
    real(8), intent (in) :: th
    real(8) :: tmp
    if (kx <= 5.4d-15) then
        tmp = sin(th)
    else
        tmp = ky * (th / sin(kx))
    end if
    code = tmp
end function

Java

public static double code(double kx, double ky, double th) {
	double tmp;
	if (kx <= 5.4e-15) {
		tmp = Math.sin(th);
	} else {
		tmp = ky * (th / Math.sin(kx));
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if kx <= 5.4e-15:
		tmp = math.sin(th)
	else:
		tmp = ky * (th / math.sin(kx))
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (kx <= 5.4e-15)
		tmp = sin(th);
	else
		tmp = Float64(ky * Float64(th / sin(kx)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if (kx <= 5.4e-15)
		tmp = sin(th);
	else
		tmp = ky * (th / sin(kx));
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[kx, 5.4e-15], N[Sin[th], $MachinePrecision], N[(ky * N[(th / N[Sin[kx], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if kx < 5.40000000000000018e-15

   1. Initial program 92.0%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Step-by-step derivation

      1. associate-*l/ N/A

         \[\leadsto \frac{\sin ky \cdot \sin th}{\color{blue}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}} \]

      2. associate-/l* N/A

         \[\leadsto \sin ky \cdot \color{blue}{\frac{\sin th}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\sin ky, \color{blue}{\left(\frac{\sin th}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}\right)}\right) \]

      4. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \left(\frac{\color{blue}{\sin th}}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}\right)\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\sin th, \color{blue}{\left(\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}\right)}\right)\right) \]

      6. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\mathsf{sin.f64}\left(th\right), \left(\sqrt{\color{blue}{{\sin kx}^{2} + {\sin ky}^{2}}}\right)\right)\right) \]

      7. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\mathsf{sin.f64}\left(th\right), \left(\sqrt{{\sin ky}^{2} + {\sin kx}^{2}}\right)\right)\right) \]

      8. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\mathsf{sin.f64}\left(th\right), \left(\sqrt{\sin ky \cdot \sin ky + {\sin kx}^{2}}\right)\right)\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\mathsf{sin.f64}\left(th\right), \left(\sqrt{\sin ky \cdot \sin ky + \sin kx \cdot \sin kx}\right)\right)\right) \]

      10. hypot-define N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\mathsf{sin.f64}\left(th\right), \left(\mathsf{hypot}\left(\sin ky, \color{blue}{\sin kx}\right)\right)\right)\right) \]

      11. hypot-lowering-hypot.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\mathsf{sin.f64}\left(th\right), \mathsf{hypot.f64}\left(\sin ky, \color{blue}{\sin kx}\right)\right)\right) \]

      12. sin-lowering-sin.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\mathsf{sin.f64}\left(th\right), \mathsf{hypot.f64}\left(\mathsf{sin.f64}\left(ky\right), \sin \color{blue}{kx}\right)\right)\right) \]

      13. sin-lowering-sin.f64 99.6%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\mathsf{sin.f64}\left(th\right), \mathsf{hypot.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{sin.f64}\left(kx\right)\right)\right)\right) \]

   3. Simplified 99.6%

      \[\leadsto \color{blue}{\sin ky \cdot \frac{\sin th}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in kx around 0

      \[\leadsto \color{blue}{\sin th} \]
   6. Step-by-step derivation

      1. sin-lowering-sin.f64 29.8%

         \[\leadsto \mathsf{sin.f64}\left(th\right) \]

   7. Simplified 29.8%

      \[\leadsto \color{blue}{\sin th} \]

   if 5.40000000000000018e-15 < kx

   1. Initial program 99.4%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing

   3. Taylor expanded in kx around inf

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sin.f64}\left(ky\right), \color{blue}{\left(\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}\right)}\right), \mathsf{sin.f64}\left(th\right)\right) \]
   4. Step-by-step derivation

      1. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sin.f64}\left(ky\right), \left(\sqrt{\sin kx \cdot \sin kx + {\sin ky}^{2}}\right)\right), \mathsf{sin.f64}\left(th\right)\right) \]

      2. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sin.f64}\left(ky\right), \left(\sqrt{\sin kx \cdot \sin kx + \sin ky \cdot \sin ky}\right)\right), \mathsf{sin.f64}\left(th\right)\right) \]

      3. hypot-define N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sin.f64}\left(ky\right), \left(\mathsf{hypot}\left(\sin kx, \sin ky\right)\right)\right), \mathsf{sin.f64}\left(th\right)\right) \]

      4. hypot-lowering-hypot.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{hypot.f64}\left(\sin kx, \sin ky\right)\right), \mathsf{sin.f64}\left(th\right)\right) \]

      5. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{hypot.f64}\left(\mathsf{sin.f64}\left(kx\right), \sin ky\right)\right), \mathsf{sin.f64}\left(th\right)\right) \]

      6. sin-lowering-sin.f64 99.4%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{hypot.f64}\left(\mathsf{sin.f64}\left(kx\right), \mathsf{sin.f64}\left(ky\right)\right)\right), \mathsf{sin.f64}\left(th\right)\right) \]

   5. Simplified 99.4%

      \[\leadsto \frac{\sin ky}{\color{blue}{\mathsf{hypot}\left(\sin kx, \sin ky\right)}} \cdot \sin th \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \sin th \cdot \color{blue}{\frac{\sin ky}{\sqrt{\sin kx \cdot \sin kx + \sin ky \cdot \sin ky}}} \]

      2. clear-num N/A

         \[\leadsto \sin th \cdot \frac{1}{\color{blue}{\frac{\sqrt{\sin kx \cdot \sin kx + \sin ky \cdot \sin ky}}{\sin ky}}} \]

      3. un-div-inv N/A

         \[\leadsto \frac{\sin th}{\color{blue}{\frac{\sqrt{\sin kx \cdot \sin kx + \sin ky \cdot \sin ky}}{\sin ky}}} \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\sin th, \color{blue}{\left(\frac{\sqrt{\sin kx \cdot \sin kx + \sin ky \cdot \sin ky}}{\sin ky}\right)}\right) \]

      5. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sin.f64}\left(th\right), \left(\frac{\color{blue}{\sqrt{\sin kx \cdot \sin kx + \sin ky \cdot \sin ky}}}{\sin ky}\right)\right) \]

      6. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sin.f64}\left(th\right), \left(\frac{\sqrt{\sin ky \cdot \sin ky + \sin kx \cdot \sin kx}}{\sin ky}\right)\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sin.f64}\left(th\right), \mathsf{/.f64}\left(\left(\sqrt{\sin ky \cdot \sin ky + \sin kx \cdot \sin kx}\right), \color{blue}{\sin ky}\right)\right) \]

      8. hypot-define N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sin.f64}\left(th\right), \mathsf{/.f64}\left(\left(\mathsf{hypot}\left(\sin ky, \sin kx\right)\right), \sin \color{blue}{ky}\right)\right) \]

      9. hypot-lowering-hypot.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sin.f64}\left(th\right), \mathsf{/.f64}\left(\mathsf{hypot.f64}\left(\sin ky, \sin kx\right), \sin \color{blue}{ky}\right)\right) \]

      10. sin-lowering-sin.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{sin.f64}\left(th\right), \mathsf{/.f64}\left(\mathsf{hypot.f64}\left(\mathsf{sin.f64}\left(ky\right), \sin kx\right), \sin ky\right)\right) \]

      11. sin-lowering-sin.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{sin.f64}\left(th\right), \mathsf{/.f64}\left(\mathsf{hypot.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{sin.f64}\left(kx\right)\right), \sin ky\right)\right) \]

      12. sin-lowering-sin.f64 99.4%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{sin.f64}\left(th\right), \mathsf{/.f64}\left(\mathsf{hypot.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{sin.f64}\left(kx\right)\right), \mathsf{sin.f64}\left(ky\right)\right)\right) \]

   7. Applied egg-rr 99.4%

      \[\leadsto \color{blue}{\frac{\sin th}{\frac{\mathsf{hypot}\left(\sin ky, \sin kx\right)}{\sin ky}}} \]

   8. Taylor expanded in ky around 0

      \[\leadsto \mathsf{/.f64}\left(\mathsf{sin.f64}\left(th\right), \color{blue}{\left(\frac{\sin kx}{ky}\right)}\right) \]
   9. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sin.f64}\left(th\right), \mathsf{/.f64}\left(\sin kx, \color{blue}{ky}\right)\right) \]

      2. sin-lowering-sin.f64 32.7%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sin.f64}\left(th\right), \mathsf{/.f64}\left(\mathsf{sin.f64}\left(kx\right), ky\right)\right) \]

   10. Simplified 32.7%

       \[\leadsto \frac{\sin th}{\color{blue}{\frac{\sin kx}{ky}}} \]

   11. Taylor expanded in th around 0

       \[\leadsto \color{blue}{\frac{ky \cdot th}{\sin kx}} \]
   12. Step-by-step derivation

       1. associate-/l* N/A

          \[\leadsto ky \cdot \color{blue}{\frac{th}{\sin kx}} \]

       2. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(ky, \color{blue}{\left(\frac{th}{\sin kx}\right)}\right) \]

       3. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(ky, \mathsf{/.f64}\left(th, \color{blue}{\sin kx}\right)\right) \]

       4. sin-lowering-sin.f64 22.7%

          \[\leadsto \mathsf{*.f64}\left(ky, \mathsf{/.f64}\left(th, \mathsf{sin.f64}\left(kx\right)\right)\right) \]

   13. Simplified 22.7%

       \[\leadsto \color{blue}{ky \cdot \frac{th}{\sin kx}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 20: 25.7% accurate, 6.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;kx \leq 6.8 \cdot 10^{-15}:\\ \;\;\;\;\sin th\\ \mathbf{else}:\\ \;\;\;\;ky \cdot \frac{th}{kx}\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= kx 6.8e-15) (sin th) (* ky (/ th kx))))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (kx <= 6.8e-15) {
		tmp = sin(th);
	} else {
		tmp = ky * (th / kx);
	}
	return tmp;
}

Fortran

real(8) function code(kx, ky, th)
    real(8), intent (in) :: kx
    real(8), intent (in) :: ky
    real(8), intent (in) :: th
    real(8) :: tmp
    if (kx <= 6.8d-15) then
        tmp = sin(th)
    else
        tmp = ky * (th / kx)
    end if
    code = tmp
end function

Java

public static double code(double kx, double ky, double th) {
	double tmp;
	if (kx <= 6.8e-15) {
		tmp = Math.sin(th);
	} else {
		tmp = ky * (th / kx);
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if kx <= 6.8e-15:
		tmp = math.sin(th)
	else:
		tmp = ky * (th / kx)
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (kx <= 6.8e-15)
		tmp = sin(th);
	else
		tmp = Float64(ky * Float64(th / kx));
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if (kx <= 6.8e-15)
		tmp = sin(th);
	else
		tmp = ky * (th / kx);
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[kx, 6.8e-15], N[Sin[th], $MachinePrecision], N[(ky * N[(th / kx), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if kx < 6.8000000000000001e-15

   1. Initial program 92.0%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Step-by-step derivation

      1. associate-*l/ N/A

         \[\leadsto \frac{\sin ky \cdot \sin th}{\color{blue}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}} \]

      2. associate-/l* N/A

         \[\leadsto \sin ky \cdot \color{blue}{\frac{\sin th}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\sin ky, \color{blue}{\left(\frac{\sin th}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}\right)}\right) \]

      4. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \left(\frac{\color{blue}{\sin th}}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}\right)\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\sin th, \color{blue}{\left(\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}\right)}\right)\right) \]

      6. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\mathsf{sin.f64}\left(th\right), \left(\sqrt{\color{blue}{{\sin kx}^{2} + {\sin ky}^{2}}}\right)\right)\right) \]

      7. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\mathsf{sin.f64}\left(th\right), \left(\sqrt{{\sin ky}^{2} + {\sin kx}^{2}}\right)\right)\right) \]

      8. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\mathsf{sin.f64}\left(th\right), \left(\sqrt{\sin ky \cdot \sin ky + {\sin kx}^{2}}\right)\right)\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\mathsf{sin.f64}\left(th\right), \left(\sqrt{\sin ky \cdot \sin ky + \sin kx \cdot \sin kx}\right)\right)\right) \]

      10. hypot-define N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\mathsf{sin.f64}\left(th\right), \left(\mathsf{hypot}\left(\sin ky, \color{blue}{\sin kx}\right)\right)\right)\right) \]

      11. hypot-lowering-hypot.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\mathsf{sin.f64}\left(th\right), \mathsf{hypot.f64}\left(\sin ky, \color{blue}{\sin kx}\right)\right)\right) \]

      12. sin-lowering-sin.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\mathsf{sin.f64}\left(th\right), \mathsf{hypot.f64}\left(\mathsf{sin.f64}\left(ky\right), \sin \color{blue}{kx}\right)\right)\right) \]

      13. sin-lowering-sin.f64 99.6%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\mathsf{sin.f64}\left(th\right), \mathsf{hypot.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{sin.f64}\left(kx\right)\right)\right)\right) \]

   3. Simplified 99.6%

      \[\leadsto \color{blue}{\sin ky \cdot \frac{\sin th}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in kx around 0

      \[\leadsto \color{blue}{\sin th} \]
   6. Step-by-step derivation

      1. sin-lowering-sin.f64 29.8%

         \[\leadsto \mathsf{sin.f64}\left(th\right) \]

   7. Simplified 29.8%

      \[\leadsto \color{blue}{\sin th} \]

   if 6.8000000000000001e-15 < kx

   1. Initial program 99.4%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Step-by-step derivation

      1. associate-*l/ N/A

         \[\leadsto \frac{\sin ky \cdot \sin th}{\color{blue}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}} \]

      2. associate-/l* N/A

         \[\leadsto \sin ky \cdot \color{blue}{\frac{\sin th}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\sin ky, \color{blue}{\left(\frac{\sin th}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}\right)}\right) \]

      4. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \left(\frac{\color{blue}{\sin th}}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}\right)\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\sin th, \color{blue}{\left(\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}\right)}\right)\right) \]

      6. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\mathsf{sin.f64}\left(th\right), \left(\sqrt{\color{blue}{{\sin kx}^{2} + {\sin ky}^{2}}}\right)\right)\right) \]

      7. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\mathsf{sin.f64}\left(th\right), \left(\sqrt{{\sin ky}^{2} + {\sin kx}^{2}}\right)\right)\right) \]

      8. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\mathsf{sin.f64}\left(th\right), \left(\sqrt{\sin ky \cdot \sin ky + {\sin kx}^{2}}\right)\right)\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\mathsf{sin.f64}\left(th\right), \left(\sqrt{\sin ky \cdot \sin ky + \sin kx \cdot \sin kx}\right)\right)\right) \]

      10. hypot-define N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\mathsf{sin.f64}\left(th\right), \left(\mathsf{hypot}\left(\sin ky, \color{blue}{\sin kx}\right)\right)\right)\right) \]

      11. hypot-lowering-hypot.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\mathsf{sin.f64}\left(th\right), \mathsf{hypot.f64}\left(\sin ky, \color{blue}{\sin kx}\right)\right)\right) \]

      12. sin-lowering-sin.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\mathsf{sin.f64}\left(th\right), \mathsf{hypot.f64}\left(\mathsf{sin.f64}\left(ky\right), \sin \color{blue}{kx}\right)\right)\right) \]

      13. sin-lowering-sin.f64 99.4%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\mathsf{sin.f64}\left(th\right), \mathsf{hypot.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{sin.f64}\left(kx\right)\right)\right)\right) \]

   3. Simplified 99.4%

      \[\leadsto \color{blue}{\sin ky \cdot \frac{\sin th}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in th around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\color{blue}{\left(th \cdot \left(1 + {th}^{2} \cdot \left(\frac{1}{120} \cdot {th}^{2} - \frac{1}{6}\right)\right)\right)}, \mathsf{hypot.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{sin.f64}\left(kx\right)\right)\right)\right) \]
   6. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(th, \left(1 + {th}^{2} \cdot \left(\frac{1}{120} \cdot {th}^{2} - \frac{1}{6}\right)\right)\right), \mathsf{hypot.f64}\left(\color{blue}{\mathsf{sin.f64}\left(ky\right)}, \mathsf{sin.f64}\left(kx\right)\right)\right)\right) \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(th, \mathsf{+.f64}\left(1, \left({th}^{2} \cdot \left(\frac{1}{120} \cdot {th}^{2} - \frac{1}{6}\right)\right)\right)\right), \mathsf{hypot.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{sin.f64}\left(kx\right)\right)\right)\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(th, \mathsf{+.f64}\left(1, \left(\left(th \cdot th\right) \cdot \left(\frac{1}{120} \cdot {th}^{2} - \frac{1}{6}\right)\right)\right)\right), \mathsf{hypot.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{sin.f64}\left(kx\right)\right)\right)\right) \]

      4. associate-*l* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(th, \mathsf{+.f64}\left(1, \left(th \cdot \left(th \cdot \left(\frac{1}{120} \cdot {th}^{2} - \frac{1}{6}\right)\right)\right)\right)\right), \mathsf{hypot.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{sin.f64}\left(kx\right)\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(th, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(th, \left(th \cdot \left(\frac{1}{120} \cdot {th}^{2} - \frac{1}{6}\right)\right)\right)\right)\right), \mathsf{hypot.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{sin.f64}\left(kx\right)\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(th, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(th, \mathsf{*.f64}\left(th, \left(\frac{1}{120} \cdot {th}^{2} - \frac{1}{6}\right)\right)\right)\right)\right), \mathsf{hypot.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{sin.f64}\left(kx\right)\right)\right)\right) \]

      7. sub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(th, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(th, \mathsf{*.f64}\left(th, \left(\frac{1}{120} \cdot {th}^{2} + \left(\mathsf{neg}\left(\frac{1}{6}\right)\right)\right)\right)\right)\right)\right), \mathsf{hypot.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{sin.f64}\left(kx\right)\right)\right)\right) \]

      8. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(th, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(th, \mathsf{*.f64}\left(th, \left(\frac{1}{120} \cdot {th}^{2} + \frac{-1}{6}\right)\right)\right)\right)\right), \mathsf{hypot.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{sin.f64}\left(kx\right)\right)\right)\right) \]

      9. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(th, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(th, \mathsf{*.f64}\left(th, \left(\frac{-1}{6} + \frac{1}{120} \cdot {th}^{2}\right)\right)\right)\right)\right), \mathsf{hypot.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{sin.f64}\left(kx\right)\right)\right)\right) \]

      10. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(th, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(th, \mathsf{*.f64}\left(th, \mathsf{+.f64}\left(\frac{-1}{6}, \left(\frac{1}{120} \cdot {th}^{2}\right)\right)\right)\right)\right)\right), \mathsf{hypot.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{sin.f64}\left(kx\right)\right)\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(th, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(th, \mathsf{*.f64}\left(th, \mathsf{+.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(\frac{1}{120}, \left({th}^{2}\right)\right)\right)\right)\right)\right)\right), \mathsf{hypot.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{sin.f64}\left(kx\right)\right)\right)\right) \]

      12. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(th, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(th, \mathsf{*.f64}\left(th, \mathsf{+.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(\frac{1}{120}, \left(th \cdot th\right)\right)\right)\right)\right)\right)\right), \mathsf{hypot.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{sin.f64}\left(kx\right)\right)\right)\right) \]

      13. *-lowering-*.f64 58.9%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(th, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(th, \mathsf{*.f64}\left(th, \mathsf{+.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(\frac{1}{120}, \mathsf{*.f64}\left(th, th\right)\right)\right)\right)\right)\right)\right), \mathsf{hypot.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{sin.f64}\left(kx\right)\right)\right)\right) \]

   7. Simplified 58.9%

      \[\leadsto \sin ky \cdot \frac{\color{blue}{th \cdot \left(1 + th \cdot \left(th \cdot \left(-0.16666666666666666 + 0.008333333333333333 \cdot \left(th \cdot th\right)\right)\right)\right)}}{\mathsf{hypot}\left(\sin ky, \sin kx\right)} \]

   8. Taylor expanded in kx around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(th, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(th, \mathsf{*.f64}\left(th, \mathsf{+.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(\frac{1}{120}, \mathsf{*.f64}\left(th, th\right)\right)\right)\right)\right)\right)\right), \mathsf{hypot.f64}\left(\mathsf{sin.f64}\left(ky\right), \color{blue}{kx}\right)\right)\right) \]
   9. Step-by-step derivation

   10. Simplified 19.3%

       \[\leadsto \sin ky \cdot \frac{th \cdot \left(1 + th \cdot \left(th \cdot \left(-0.16666666666666666 + 0.008333333333333333 \cdot \left(th \cdot th\right)\right)\right)\right)}{\mathsf{hypot}\left(\sin ky, \color{blue}{kx}\right)} \]

   11. Taylor expanded in th around 0

       \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\color{blue}{th}, \mathsf{hypot.f64}\left(\mathsf{sin.f64}\left(ky\right), kx\right)\right)\right) \]
   12. Step-by-step derivation

   13. Simplified 19.5%

       \[\leadsto \sin ky \cdot \frac{\color{blue}{th}}{\mathsf{hypot}\left(\sin ky, kx\right)} \]

   14. Taylor expanded in ky around 0

       \[\leadsto \color{blue}{\frac{ky \cdot th}{kx}} \]
   15. Step-by-step derivation

       1. associate-/l* N/A

          \[\leadsto ky \cdot \color{blue}{\frac{th}{kx}} \]

       2. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(ky, \color{blue}{\left(\frac{th}{kx}\right)}\right) \]

       3. /-lowering-/.f64 18.0%

          \[\leadsto \mathsf{*.f64}\left(ky, \mathsf{/.f64}\left(th, \color{blue}{kx}\right)\right) \]

   16. Simplified 18.0%

       \[\leadsto \color{blue}{ky \cdot \frac{th}{kx}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 21: 17.8% accurate, 70.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;ky \leq 5.8 \cdot 10^{-96}:\\ \;\;\;\;ky \cdot \frac{th}{kx}\\ \mathbf{else}:\\ \;\;\;\;th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= ky 5.8e-96) (* ky (/ th kx)) th))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (ky <= 5.8e-96) {
		tmp = ky * (th / kx);
	} else {
		tmp = th;
	}
	return tmp;
}

Fortran

real(8) function code(kx, ky, th)
    real(8), intent (in) :: kx
    real(8), intent (in) :: ky
    real(8), intent (in) :: th
    real(8) :: tmp
    if (ky <= 5.8d-96) then
        tmp = ky * (th / kx)
    else
        tmp = th
    end if
    code = tmp
end function

Java

public static double code(double kx, double ky, double th) {
	double tmp;
	if (ky <= 5.8e-96) {
		tmp = ky * (th / kx);
	} else {
		tmp = th;
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if ky <= 5.8e-96:
		tmp = ky * (th / kx)
	else:
		tmp = th
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (ky <= 5.8e-96)
		tmp = Float64(ky * Float64(th / kx));
	else
		tmp = th;
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if (ky <= 5.8e-96)
		tmp = ky * (th / kx);
	else
		tmp = th;
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[ky, 5.8e-96], N[(ky * N[(th / kx), $MachinePrecision]), $MachinePrecision], th]

Derivation

1. Split input into 2 regimes

2. if ky < 5.79999999999999987e-96

   1. Initial program 91.2%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Step-by-step derivation

      1. associate-*l/ N/A

         \[\leadsto \frac{\sin ky \cdot \sin th}{\color{blue}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}} \]

      2. associate-/l* N/A

         \[\leadsto \sin ky \cdot \color{blue}{\frac{\sin th}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\sin ky, \color{blue}{\left(\frac{\sin th}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}\right)}\right) \]

      4. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \left(\frac{\color{blue}{\sin th}}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}\right)\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\sin th, \color{blue}{\left(\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}\right)}\right)\right) \]

      6. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\mathsf{sin.f64}\left(th\right), \left(\sqrt{\color{blue}{{\sin kx}^{2} + {\sin ky}^{2}}}\right)\right)\right) \]

      7. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\mathsf{sin.f64}\left(th\right), \left(\sqrt{{\sin ky}^{2} + {\sin kx}^{2}}\right)\right)\right) \]

      8. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\mathsf{sin.f64}\left(th\right), \left(\sqrt{\sin ky \cdot \sin ky + {\sin kx}^{2}}\right)\right)\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\mathsf{sin.f64}\left(th\right), \left(\sqrt{\sin ky \cdot \sin ky + \sin kx \cdot \sin kx}\right)\right)\right) \]

      10. hypot-define N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\mathsf{sin.f64}\left(th\right), \left(\mathsf{hypot}\left(\sin ky, \color{blue}{\sin kx}\right)\right)\right)\right) \]

      11. hypot-lowering-hypot.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\mathsf{sin.f64}\left(th\right), \mathsf{hypot.f64}\left(\sin ky, \color{blue}{\sin kx}\right)\right)\right) \]

      12. sin-lowering-sin.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\mathsf{sin.f64}\left(th\right), \mathsf{hypot.f64}\left(\mathsf{sin.f64}\left(ky\right), \sin \color{blue}{kx}\right)\right)\right) \]

      13. sin-lowering-sin.f64 99.6%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\mathsf{sin.f64}\left(th\right), \mathsf{hypot.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{sin.f64}\left(kx\right)\right)\right)\right) \]

   3. Simplified 99.6%

      \[\leadsto \color{blue}{\sin ky \cdot \frac{\sin th}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in th around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\color{blue}{\left(th \cdot \left(1 + {th}^{2} \cdot \left(\frac{1}{120} \cdot {th}^{2} - \frac{1}{6}\right)\right)\right)}, \mathsf{hypot.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{sin.f64}\left(kx\right)\right)\right)\right) \]
   6. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(th, \left(1 + {th}^{2} \cdot \left(\frac{1}{120} \cdot {th}^{2} - \frac{1}{6}\right)\right)\right), \mathsf{hypot.f64}\left(\color{blue}{\mathsf{sin.f64}\left(ky\right)}, \mathsf{sin.f64}\left(kx\right)\right)\right)\right) \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(th, \mathsf{+.f64}\left(1, \left({th}^{2} \cdot \left(\frac{1}{120} \cdot {th}^{2} - \frac{1}{6}\right)\right)\right)\right), \mathsf{hypot.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{sin.f64}\left(kx\right)\right)\right)\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(th, \mathsf{+.f64}\left(1, \left(\left(th \cdot th\right) \cdot \left(\frac{1}{120} \cdot {th}^{2} - \frac{1}{6}\right)\right)\right)\right), \mathsf{hypot.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{sin.f64}\left(kx\right)\right)\right)\right) \]

      4. associate-*l* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(th, \mathsf{+.f64}\left(1, \left(th \cdot \left(th \cdot \left(\frac{1}{120} \cdot {th}^{2} - \frac{1}{6}\right)\right)\right)\right)\right), \mathsf{hypot.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{sin.f64}\left(kx\right)\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(th, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(th, \left(th \cdot \left(\frac{1}{120} \cdot {th}^{2} - \frac{1}{6}\right)\right)\right)\right)\right), \mathsf{hypot.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{sin.f64}\left(kx\right)\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(th, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(th, \mathsf{*.f64}\left(th, \left(\frac{1}{120} \cdot {th}^{2} - \frac{1}{6}\right)\right)\right)\right)\right), \mathsf{hypot.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{sin.f64}\left(kx\right)\right)\right)\right) \]

      7. sub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(th, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(th, \mathsf{*.f64}\left(th, \left(\frac{1}{120} \cdot {th}^{2} + \left(\mathsf{neg}\left(\frac{1}{6}\right)\right)\right)\right)\right)\right)\right), \mathsf{hypot.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{sin.f64}\left(kx\right)\right)\right)\right) \]

      8. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(th, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(th, \mathsf{*.f64}\left(th, \left(\frac{1}{120} \cdot {th}^{2} + \frac{-1}{6}\right)\right)\right)\right)\right), \mathsf{hypot.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{sin.f64}\left(kx\right)\right)\right)\right) \]

      9. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(th, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(th, \mathsf{*.f64}\left(th, \left(\frac{-1}{6} + \frac{1}{120} \cdot {th}^{2}\right)\right)\right)\right)\right), \mathsf{hypot.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{sin.f64}\left(kx\right)\right)\right)\right) \]

      10. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(th, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(th, \mathsf{*.f64}\left(th, \mathsf{+.f64}\left(\frac{-1}{6}, \left(\frac{1}{120} \cdot {th}^{2}\right)\right)\right)\right)\right)\right), \mathsf{hypot.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{sin.f64}\left(kx\right)\right)\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(th, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(th, \mathsf{*.f64}\left(th, \mathsf{+.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(\frac{1}{120}, \left({th}^{2}\right)\right)\right)\right)\right)\right)\right), \mathsf{hypot.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{sin.f64}\left(kx\right)\right)\right)\right) \]

      12. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(th, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(th, \mathsf{*.f64}\left(th, \mathsf{+.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(\frac{1}{120}, \left(th \cdot th\right)\right)\right)\right)\right)\right)\right), \mathsf{hypot.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{sin.f64}\left(kx\right)\right)\right)\right) \]

      13. *-lowering-*.f64 47.3%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(th, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(th, \mathsf{*.f64}\left(th, \mathsf{+.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(\frac{1}{120}, \mathsf{*.f64}\left(th, th\right)\right)\right)\right)\right)\right)\right), \mathsf{hypot.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{sin.f64}\left(kx\right)\right)\right)\right) \]

   7. Simplified 47.3%

      \[\leadsto \sin ky \cdot \frac{\color{blue}{th \cdot \left(1 + th \cdot \left(th \cdot \left(-0.16666666666666666 + 0.008333333333333333 \cdot \left(th \cdot th\right)\right)\right)\right)}}{\mathsf{hypot}\left(\sin ky, \sin kx\right)} \]

   8. Taylor expanded in kx around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(th, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(th, \mathsf{*.f64}\left(th, \mathsf{+.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(\frac{1}{120}, \mathsf{*.f64}\left(th, th\right)\right)\right)\right)\right)\right)\right), \mathsf{hypot.f64}\left(\mathsf{sin.f64}\left(ky\right), \color{blue}{kx}\right)\right)\right) \]
   9. Step-by-step derivation

   10. Simplified 31.2%

       \[\leadsto \sin ky \cdot \frac{th \cdot \left(1 + th \cdot \left(th \cdot \left(-0.16666666666666666 + 0.008333333333333333 \cdot \left(th \cdot th\right)\right)\right)\right)}{\mathsf{hypot}\left(\sin ky, \color{blue}{kx}\right)} \]

   11. Taylor expanded in th around 0

       \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\color{blue}{th}, \mathsf{hypot.f64}\left(\mathsf{sin.f64}\left(ky\right), kx\right)\right)\right) \]
   12. Step-by-step derivation

   13. Simplified 31.8%

       \[\leadsto \sin ky \cdot \frac{\color{blue}{th}}{\mathsf{hypot}\left(\sin ky, kx\right)} \]

   14. Taylor expanded in ky around 0

       \[\leadsto \color{blue}{\frac{ky \cdot th}{kx}} \]
   15. Step-by-step derivation

       1. associate-/l* N/A

          \[\leadsto ky \cdot \color{blue}{\frac{th}{kx}} \]

       2. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(ky, \color{blue}{\left(\frac{th}{kx}\right)}\right) \]

       3. /-lowering-/.f64 14.8%

          \[\leadsto \mathsf{*.f64}\left(ky, \mathsf{/.f64}\left(th, \color{blue}{kx}\right)\right) \]

   16. Simplified 14.8%

       \[\leadsto \color{blue}{ky \cdot \frac{th}{kx}} \]

   if 5.79999999999999987e-96 < ky

   1. Initial program 99.7%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Step-by-step derivation

      1. associate-*l/ N/A

         \[\leadsto \frac{\sin ky \cdot \sin th}{\color{blue}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}} \]

      2. associate-/l* N/A

         \[\leadsto \sin ky \cdot \color{blue}{\frac{\sin th}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\sin ky, \color{blue}{\left(\frac{\sin th}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}\right)}\right) \]

      4. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \left(\frac{\color{blue}{\sin th}}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}\right)\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\sin th, \color{blue}{\left(\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}\right)}\right)\right) \]

      6. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\mathsf{sin.f64}\left(th\right), \left(\sqrt{\color{blue}{{\sin kx}^{2} + {\sin ky}^{2}}}\right)\right)\right) \]

      7. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\mathsf{sin.f64}\left(th\right), \left(\sqrt{{\sin ky}^{2} + {\sin kx}^{2}}\right)\right)\right) \]

      8. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\mathsf{sin.f64}\left(th\right), \left(\sqrt{\sin ky \cdot \sin ky + {\sin kx}^{2}}\right)\right)\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\mathsf{sin.f64}\left(th\right), \left(\sqrt{\sin ky \cdot \sin ky + \sin kx \cdot \sin kx}\right)\right)\right) \]

      10. hypot-define N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\mathsf{sin.f64}\left(th\right), \left(\mathsf{hypot}\left(\sin ky, \color{blue}{\sin kx}\right)\right)\right)\right) \]

      11. hypot-lowering-hypot.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\mathsf{sin.f64}\left(th\right), \mathsf{hypot.f64}\left(\sin ky, \color{blue}{\sin kx}\right)\right)\right) \]

      12. sin-lowering-sin.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\mathsf{sin.f64}\left(th\right), \mathsf{hypot.f64}\left(\mathsf{sin.f64}\left(ky\right), \sin \color{blue}{kx}\right)\right)\right) \]

      13. sin-lowering-sin.f64 99.5%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\mathsf{sin.f64}\left(th\right), \mathsf{hypot.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{sin.f64}\left(kx\right)\right)\right)\right) \]

   3. Simplified 99.5%

      \[\leadsto \color{blue}{\sin ky \cdot \frac{\sin th}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in kx around 0

      \[\leadsto \color{blue}{\sin th} \]
   6. Step-by-step derivation

      1. sin-lowering-sin.f64 37.0%

         \[\leadsto \mathsf{sin.f64}\left(th\right) \]

   7. Simplified 37.0%

      \[\leadsto \color{blue}{\sin th} \]

   8. Taylor expanded in th around 0

      \[\leadsto \color{blue}{th} \]
   9. Step-by-step derivation

   10. Simplified 17.9%

       \[\leadsto \color{blue}{th} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 22: 13.7% accurate, 709.0× speedup

Math

\[\begin{array}{l} \\ th \end{array} \]

FPCore

(FPCore (kx ky th) :precision binary64 th)

C

double code(double kx, double ky, double th) {
	return th;
}

Fortran

real(8) function code(kx, ky, th)
    real(8), intent (in) :: kx
    real(8), intent (in) :: ky
    real(8), intent (in) :: th
    code = th
end function

Java

public static double code(double kx, double ky, double th) {
	return th;
}

Python

def code(kx, ky, th):
	return th

Julia

function code(kx, ky, th)
	return th
end

MATLAB

function tmp = code(kx, ky, th)
	tmp = th;
end

Wolfram

code[kx_, ky_, th_] := th

Derivation

1. Initial program 93.7%

   \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
2. Step-by-step derivation

   1. associate-*l/ N/A

      \[\leadsto \frac{\sin ky \cdot \sin th}{\color{blue}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}} \]

   2. associate-/l* N/A

      \[\leadsto \sin ky \cdot \color{blue}{\frac{\sin th}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}} \]

   3. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\sin ky, \color{blue}{\left(\frac{\sin th}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}\right)}\right) \]

   4. sin-lowering-sin.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \left(\frac{\color{blue}{\sin th}}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}\right)\right) \]

   5. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\sin th, \color{blue}{\left(\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}\right)}\right)\right) \]

   6. sin-lowering-sin.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\mathsf{sin.f64}\left(th\right), \left(\sqrt{\color{blue}{{\sin kx}^{2} + {\sin ky}^{2}}}\right)\right)\right) \]

   7. +-commutative N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\mathsf{sin.f64}\left(th\right), \left(\sqrt{{\sin ky}^{2} + {\sin kx}^{2}}\right)\right)\right) \]

   8. unpow2 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\mathsf{sin.f64}\left(th\right), \left(\sqrt{\sin ky \cdot \sin ky + {\sin kx}^{2}}\right)\right)\right) \]

   9. unpow2 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\mathsf{sin.f64}\left(th\right), \left(\sqrt{\sin ky \cdot \sin ky + \sin kx \cdot \sin kx}\right)\right)\right) \]

   10. hypot-define N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\mathsf{sin.f64}\left(th\right), \left(\mathsf{hypot}\left(\sin ky, \color{blue}{\sin kx}\right)\right)\right)\right) \]

   11. hypot-lowering-hypot.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\mathsf{sin.f64}\left(th\right), \mathsf{hypot.f64}\left(\sin ky, \color{blue}{\sin kx}\right)\right)\right) \]

   12. sin-lowering-sin.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\mathsf{sin.f64}\left(th\right), \mathsf{hypot.f64}\left(\mathsf{sin.f64}\left(ky\right), \sin \color{blue}{kx}\right)\right)\right) \]

   13. sin-lowering-sin.f64 99.6%

       \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\mathsf{sin.f64}\left(th\right), \mathsf{hypot.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{sin.f64}\left(kx\right)\right)\right)\right) \]

3. Simplified 99.6%

   \[\leadsto \color{blue}{\sin ky \cdot \frac{\sin th}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}} \]
4. Add Preprocessing

5. Taylor expanded in kx around 0

   \[\leadsto \color{blue}{\sin th} \]
6. Step-by-step derivation

   1. sin-lowering-sin.f64 24.6%

      \[\leadsto \mathsf{sin.f64}\left(th\right) \]

7. Simplified 24.6%

   \[\leadsto \color{blue}{\sin th} \]

8. Taylor expanded in th around 0

   \[\leadsto \color{blue}{th} \]
9. Step-by-step derivation

10. Simplified 13.2%

    \[\leadsto \color{blue}{th} \]
11. Add Preprocessing

Reproduce

herbie shell --seed 2024150 
(FPCore (kx ky th)
  :name "Toniolo and Linder, Equation (3b), real"
  :precision binary64
  (* (/ (sin ky) (sqrt (+ (pow (sin kx) 2.0) (pow (sin ky) 2.0)))) (sin th)))